A treatise on astronomy

Presented to the

LIBRARY of the
UNIVERSITY OF TORONTO

by

Miss M. Robertson

TREATISE ON ASTRONOJVIY.

BY

SIR JOHN F. W. HERSCHEL, KNT. GUELP.

F.R.S.L. &E. M.R.I.A. F.R.A.S. F.G.S. M.C.U.P.S.

COEHESrONDEJJT OF THE ROYAL ACADEMY, OF SCIENCES OF PARIS, AND 0THE8
FOREIGN SCIENTIFIC INSTITUTIONS.

THIRD EDITION.

PHILADELPHIA.
CAREY, LEA & BLANCHARD.

1835.

^--V ^r^

" Et quoniam eadem natura cupiditatem ingenult hominibus veri
inveniendi, quod facillimi apparet, cum vacui curis, etiam quid in
coelo fiat, scire avemus : his initiis inducti omnia vera diligimus ; id
est, fidelia, simplicia, constantia ; turn vana, falsa, fallendia odimus."

Cicero, de Fin. Jion. ct jMcil. ii. 14.

And forasmuch as nature itself has implanted in man a craving
after the discovery of truth (which appears most clearly from this,
that, when unoppressed by cares, we delight to know even what is
going on in the heavens), — led by this instinct, we learn to love all
truth for its own sake ; that is to say, whatever is faithful, simple,
and consistent ; while we hold in abhorrence whatever is empty,
deceptive, or untrue.

2

CONTENTS.

Introduction Page 7

CHAPTER I.

General Notions — Form and Magnitude of the Earth — Horizon and
its Dip — Tlie Atinos])here — Refrartion — Twilight — Appearances
resuhing from diurnal Motion — Parallax — First Step towards form-
ing an Idea of the Distance of the Stars — Definitions - - - 14

CHAPTER II.

Of the Nature of astronomical Instruments and Observations in gene-
ral— Of sidereal and solar Time — Of the Measurement of Time —
Clocks, Chronometers, the Transit Instrument — Of the Measure-
ment of angular Intervals — Application of the Telescope to Instru-
ments destined to tliat Purpose — Of the Mural Circle — Determina-
tion of polar and horizontal Points — Tlie Level — Plumb Line —
Artificial Horizon — Collimator — Of compound Instruments with
co-ordinate Circles, the Equatorial — Altitude and Azimuth Instru-
ment— Of the Sextant and Reflecting Circle — Principle of Repeti-
tion 66

CHAPTER III.

OF GEOGRAPHY.

Of the Figure of the Earth — Its exact Dimensions — Its Form that of
Equilibrium mothlied by Centrifugal Force — Variation of Cravity
on its Surface — Statical and dynamical Measures of Gravity — The
Pendulum — Gravity to a Spheroid — Other Effects of Earth's Rota-
tion— Trade-winds — Determination of geographical Positions — Of
Latitudes — Of Longitudes — Conduct of a trignometrical Survey —
Of Maps — Projections of the Sphere — Measurement of Heights by
the Barometer -.-..-.... 105

CHAPTER IV.

OF URANOGRArHY.

Construction of celestial Maps and Glolx^s by Observations of right
Ascension and Declination — Celestial Objects distinguished into
fixed and erratic — Of the Constellations — Natural Regions in the
Heavens— The Milky Way — The Zodiac— Of the Ecliptic— Celes-
tial Latitudes and Longitudes — Precession of the Equinoxes — Nu-
tation— Aberration — Uranographical Problems .... 151

3

4 CONTENTS.

CHAPTER V.

OF THE sun's motion.

Page

Apparent Motion of the Sun not unifonn — Its apparent Diameter also
variable — Variation of its Distance concluded — Its apparent Orbit
an Ellipse about the Focus — Law of tlie angular Velocity — Equa-
ble Description of Areas — Parallax of the Sun — Its Distance and
Magnitude — Coperniean Explanation of the Sun's apparent Motion
— Parallelism of the Earth's Axis — The Seasons — Heat received
from the Sun in diflerent Parts of the Orbit 17€

CHAPTER VI.

Of the Moon — Its sidereal Period — Its apparent Diameter — Its Paral-
lax, Distance, and real Diameter — First Approximation to its Orbit
— An Ellipse about the Earth in the Focus — Its Eccentricity and
Inclination — Motion of the Nodes of its Orbit — Occultations — Solar
Eclipses — Pliases of the Moon — Its synodical Period — Lunar
Eclipses — Motion of the Apsides of its Orbit — Physical Constitution
of the Moon — Its Mountains — Atmosphere — EJotation on Axis —
Libration — Appearance of the Earth from it - - - 203

CHAPTER VII.

Of terrestrial Gravity — Of the Law of universal Gravitation — Paths
of Projectiles ; apparent, real — The Moon retained in licr Orbit by
Gravity — Its Law of Diminution — Laws of elliptic Motion — Orbit
of the Earth round the Sun in accordance with these Laws —
Masses of the Earth and Sun compared — Density of the Sun —
Force of Gravity at its Surface — Disturbing Eflect of the Sun on
the Moon's Motion 221

CHAPTER VIII.

OF THE SOLAR SYSTEM.

Apparent Motions of the Planets — Their Stations and Retrograda-
tinns — ^The Sun their natural Centre of Motion — Inferior Planets
— Their Phases, Periods, &c. — Dimensions and Form of their Orbits
— Transits across tlie Sun — Superior Planets, their Distances, Pe-
riods, &c. — Kepler's Laws and their Interpretation — Elliptic Ele-
ments of a Planet's Orbit — Its heliocentric and geocentric Place —
Bode's Law of Planetary Distances — The four ultra-zodiacal Pla-
nets— Physical Peculiarities observable in each of the Planets - 231

CHAPTER IX.

OF THE SATELLITES.

Of the Moon, as a Satellite of the Earth — General Proximity of Satel-
lites to their Primaries, and consefjuent Subordination of their
Motions — Masses of the Primaries concluded from the Periods of
their Satellites — Maintenance of Kepler's Laws in the secondary
Systems — Of Jupiter's Satellites — Their Eclipses, &c. — Velocity
of Light discovered by their Moans — Satellites of Saturn — Of
Uranus 272

CONTENTS. 5

CHAPTER X.

OF COMETS.

Page
Great Number of reoorded Comets— The number of unrecorded
probably much greater — Description of a Comet — Comets without
Tails — Increase and Decay ol' their Tails — Their Motions — Sub-
ject to the general Laws of planetary Motion — Elements of their
Orbits — Periodic Return of certain Comets — Halley's — Encke's —
Biela's — Dmiensions of Comets — Their Resistance by the Etlier,
gradual Decay, and passible Dispersion in Space .... 284

CHAPTER XI.

OF PERTURBATIONS.

Subject propotmded — Superposition of small Motions — Problem of
three Bodies — Estimation of disturbing Forces — Motion of Nodes
— Changes of Inclination — Compensation operated in a whole
Revolution of the Node — Lagrange's Theorem of the Stability of
the Inclinations — Chaniie of Obliquity of the Ecliptic — Precession
of the Equinoxes — Nutation— Theorem resjiecting forced Vibra-
tions—Of the Tides — Variation of Elements of the Planet's Orbits
—Periodic and secular — Disturbing Forces considered as tangen-
tial and radial — Effects of tangential Force — 1st, in circular Orbits ;
2d, in elii|)tic — Compensations effected — Case of near Commen-
Rurability of mean Motions — The great Inequality of Jupiter and
Saturn explained — The long Inequality of Venus and the Earth —
Lunar Variation — Effects of the radial Force — Moan Eflcct of the
Period and Dimensions of the disturbed Orbit — Variable Part of
its Elfect — Lunar Evoction — Secular Acceleration of the Moon's
Motion — Permanence of the Axes and Periods — Theory of the secu-
lar Variations of the Eccentricities and Perilielia — Motion of the
lunar Apsides — Lagrange's Theorem of the Stability of the Ec-
centricities— Nutation of the lunar Orbit — Perturbations of Jupi-
ter's Satellites 294

CHAPTER Xir.

OF SIDEREAL ASTRONOMY.

Of the Stars generally — ^Tlieir distribution into Classes according to
their apparent Magnitudes — Their apparent Distribution over the
Heavens — Of the Alilky Way — Annual Parallax — Real Distances,
probable Dimensions, and Nature of the Stars — Variable Stars —
Temporary Stars — Of double Stars — ^Their Revolution about each
other in elliptic Orbits — Extension of the Law of Gravity to such
Systems — Of coloured Stars — Proper Motion of the Sun and Stars
— Systematic Aberration and Parallax — Of compound sidereal
Systems — Clusters of Stars — Of Nebulas — Nebulous Stars — Annu-
lar and planetary Nebulae — Zodiacal Light - . - . . 349

a2

6 CONTENTS.

CHAPTER XIII.

Page

Of the Calendar 381

Synoptic Table of the Elements of the Solar System - - -389

Synoptic Table of the Elements of the Orbits of the Satellites, so far

as they are known 390

I. The Moon 390

II. Satellites of Jupiter 390

III. Satellites of Saturn 391 /

IV. Satellites of Uranus 391 ''^

Index - 393

TREATISE

ON

ASTRONOMY.

INTRODUCTION.

(1.) In entering upon any scientific pursuit, one of the
student's first endeavours ought to be, to prepare his
mind for the reception of truth, by dismissing, or at least
loosening his hold on, all such crude and hastily adopted
notions respecting the objects and relations he is about
to examine as may tend to embarrass or mislead him ;
and to strengthen liimself, by something of an effort and
a resolve, for the unprejudiced admission of any con-
clusion which shall appear to be supported by careful
observation and logical argument, even should it prove
of a nature adverse to notions he may have previously
formed for himself, or taken up, without examination,
on the credit of others. Such an effort is, in fact, a
commencement of that intellectual discipline which
forms one of the most important ends of all science.
It is the first movement of approach towards that state of
mental purity which alone can fit us for a full and steady
perception of moral beauty as well as physical adaptation.
It is the " euphrasy and rue" with which we must " purge
our sight" before we can receive and contemplate as they
are the lineaments of truth and nature.

(2.) There is no science which, more than astronomy,
stands in need of such a preparation, or draws more
largely on that intellectual liberality which is ready to
adopt whatever is demonstrated, or concede whatever is
rendered highly probable, however new and uncommon

7

\

8 A TREATISE ON ASTRONOMY.

the points of view may be in whicli objects the most
familiar may thereby become placed. Almost all its
conclusions stand in open and striking contradiction
with those of superficial and vulgar observation, and
with what appears to everyone, until he has understood
and weighed the proofs to the contrary, the most posi-
tive evidence of his senses. Thus, the earth on which
he stands, and which has served for ages as the un-
shaken foundation of the firmest stractures, either of art
or nature, is divested by the astronomer of its a'ttrilnite
of fixity, and conceived by him as turning swiftly on its
centre, and at the same time moving onwards through
space with ^reat rapidity. The sun and the moon,
which appear to untaught eyes round bodies of no very
considerable size, become enlarged in his imagination
into vast globes, — the one approacliing in magnitude to
the earth itself, the other immensely surpassing it. The
planets, which appear only as stars somewhat brighter
than the rest, are to him spacious, elaborate, and habit-
able Avorlds ; several of them vastly greater and far more
curiously furnished tlian the earth he inhabits, as there
are also others less so ; and the stars themselves, properly
so called, which to ordinary apprehension present only
lucid sparks or brilliant atoms, are to him suns of various
and transcendent glory — efiulgent centres of life and light
to myriads of unseen worlds : so that when, after dilat-
ing his thoughts to comprehend the grandeur of those
ideas his calculations have called up, and exhausting his
imagination and the powers of his language to devise
similes and metaphors illustrative of the immensity of
the scale on which his universe is constructed, he shrinks
back to his native sphere ; he finds it, in comparison, a
mere point; so lost — even in the minute system to
which it belongs — as to be invisible and unsuspected
from some of its principal and remoter members.

(3.) There is hardly anything which sets in a stronger
light the inherent power of truth over the mind of
man, when opposed by no motives of interest or passion,
than the perfect readiness with which all these conclu-
sions are assented to as soon as their evidence is clearly
apprehended, and the tenacious hold they acquire over

INTRODUCTION. 9

our belief when once admitted. In the conduct, therefore,
of this volume, we shall take it for granted that our
reader is more desirous to learn the system which it is
its object to teach as it now stands, than to raise or re-
vive objections against it ; and that, in short, he comes
to the task with a willing mind ; an assumption which
will not only save ourselves the trouble of piling argu-
ment on argument to convince the skeptical, but will
greatly facilitate his actual progress, inasmuch as he will
find it at once easier and more satisfactory to pursue from
the outset a straight and detinite path, than to be con-
stantly stepping aside, involving himself in perplexities
and circuits, which, after all, can only terminate in
finding himself compelled to adopt our road.

(4.) The method, therefore, we propose to follow is
neither strictly the analytic nor the synthetic, but rather
such a combination of both, with a leaning to the latter,
as may best suit with a didactic composition. Our object
is not to convince or refute opponents, nor to inquire,
under the semblance of an assumed ignorance, for prin-
ciples of which we are all the time in full possession — ■
but simply to teach what we know. The moderate limit
of a single volume, and the necessity of being on every
point, within that limit, rather diffuse and copious in ex-
planation, as well as the eminently matured and ascer-
tained character of the science itself, render this course
both practicable and eligible. Practicable, because there
is now no danger of any revolution in astronomy, like
those which are daily changing the features of the less
advanced sciences, supervening, ta destroy all our hypo-
theses, and throw our statements into confusion. Eligible,
because the space to be bestowed, either in combating
refuted systems, or in leading the reader forward by
slow and measured steps from the known to the un-
known, maybe more advantageously devoted to each ex-
planatory illustrations as will impress on him a familiar
and, as it were, a practical sense of the sequence of phe-
nomena, and the manner in which they are produced.
We shall not, then, reject the analytic course where it
leads more easily and directly to our objects, or in any
way fetter ourselves by a rigid adherence to method.

10 A TREATISE ON ASTRONOMY.

Writing only to l)e unilerstood, and to communicate as
imicli information in as little space as possible, consist-
ently witli its distinct and rjfeciual communication, we
can atTord to make no sacrifice to system, to form, or to
afTectation.

(5.) We shall take for granted, from the outset, the
Copcrnican system of the world ; relying on the easy,
obvious, and natural explanation it affords of all the phe-
nomena as they come to be described, to impress the
student with a sense of its truth, without either tlie form-
ality of demonstration or the superfluous tedium of
eulogy, calling to mind that important remark of Bacon :
— " Theoriarum vires, arcta et quasi se mutuo sustiaente
partium adaptatione, qua, quasi in orbem coherent, lir-
mantur;"* nor failing, however, to point out to the
reader, as occasion offers, the contrast which its superior
simplicity offers to the complication of other hypotheses.

(6.) The preliminary knowledge which it is desirable
that the student should possess, in order for the more
advantageous perusal of the following pages, consists in
the familiar practice of decimal and sexagesimal arith-
metic ; some moderate acquaintance with geometry and
trigonometry, botli plane and spherical ; the elementary
principles of mechanics ; and enough of optics to under-
stand the construction and use of the telescope, and some
other of the simpler instruments. For the acquisition of
these we may refer him to those other parts of this Cy-
clopaedia which profess to treat of the several subjects in
question. Of course, the more of such knowledge he
brings to the perusal, the easier will be his progress, and
the more complete the information gained ; but we shall
endeavour in every case, as far as it can be done with-
out a sacrifice of clearness, and of that useful brevity
which consists in the absence of prolixity and episode,
to render what we have to say as independent of other
books as possible.

(7.) After all, we must distinctly caution such of our
readers as may commence and terminate their astronomi-

* Tlie confirmation of theories relies on the compact adaptation of
their parts, hy which, like those of an arch or dome, they mutually
sustain each other, and form a coherent whole.

INTRODITCTION. 11

cal studies with the present work (though of such, — at
least in tlie latter predicament, — we trust the number will
be few), that its utmost pretension is to place them on
the threshold of this particular wing of the temple of sci-
ence, or rather on an eminence exterior to it, whence
they may obtain something like a general notion of its
structure ; or, at most, to give those who may wish to
enter, aground-plan of its accesses, and put them in pos-
session of the pass-word. Admission to its sanctuary,
and to the privileges and feelings of a votary, is only to
be gained by one means, — a sound and sufficient know-
ledge of mathematics, the great instrument of all exact in-
quiry, tvithout ivhich no man can ever make such ad-
vances in this or any other of the higher departments of
science, as can entitle him to form an independent opi-
nion on any subject of discussion within their range. It
is not without an effort that those who possess this know-
ledge can communicate on such subjects with those who
do not, and adapt their language and their illustrations to
the necessities of such an intercourse. Propositions
which to the one are almost identical, are theorems of
import and difficulty to tire other ; nor is their evidence
presented in the same way to the mind of each. In
teaching such propositions, under such circumstances,
the appeal has to be made, not to the pure and abstract rea-
son, but to the sense of analogy, — to practice and expe-
rience : principles and modes of action have to be esta-
blished, not by direct argument from acknowledged
axioms, but by bringing forward and dwelling on simple
and familiar instances in which the same principles and
the same or similar modes of action take place ; thus
erecting, as it were, in each particular case, a separate
induction, and constructing at each step a little body of
science to meet its exigencies. The diflerenee is that
of pioneering a road through an untraversed country, and
advancing at ease along a broad and beaten highway ;
that is to say, if we are determined to make ourselves
distinctly understood, and will appeal to reason at all.
As for the method of assertion, or a direct demand on
the faith of the student (though in some complex cases
indispensable, where illustrative explanation would defeat

13 A TREATISE ON ASTRONOMY.

its own end by becoming tedious and burdensome to both
parties), it is one which we shall neither adopt ourselves
nor would recommend to others.

(8.) On the other Imnd, althougli it is something- now
to abandon the road of mathematical demonstration in the
treatment of subjects susceptible of it, and teach any con-
siderable branch of science entirely or chielly by the way
of illustration and familiar parallels, it is yet not impossi-
ble that those who are already well acquainted with our
subject, and whose knowledge has been acquired by that
confessedly higher and better practice which is incompa-
tible with the avowed objects of the present work, may
yet find their account in its perusal, — for this reason, that
it is always of advantage to present any given body of
knowledge to the mind in as great a variety of different
lights as possible. It is a property of illustrations of this
kind to strike no two minds in the same manner, or with
the same force ; because no two minds are stored with
the same images, or have acquired their notions of them
by similar habits. Accordingly, it may very well hap-
pen, that a proposition, even to one best acquainted with
it, may be placed not merely in a new and uncommon,
but in a more impressive and satisfactory light by such
a course — some obscurity may be dissipated, some inward
misgiving cleared up, or even some link supplied Avhich
may lead to the perception of connexions and deductions
altogether unknown before. And the probability of this
is increased when, as in the present instance, the illustra-
tions chosen have not been studiously selected from books,
but are such as have presented themselves freely to the
author's mind as being most in harmony with his own
views ; by which, of course, he means to lay no claim
to originality in all or any of them beyond what they
may really possess.

(9.) Besides, there are cases in the application of me-
chanical principles with which the mathematical student
is but too familiar, where, when the data are before him,
and the numerical and geometrical relations of his pro-
blems all clear to his conception, — when his forces are
estimated and his lines measured, — nay, when even he has
followed up the application of his technical processes, and

INTRODUCTION. 13

fairly arrived at his conclusion, — there is still something
wanting in his mind — not in the evidence, for he has ex-
amined each link, and finds the chain complete — not in
the principles, for those he well knows are too firmly es-
tablished to be shaken — but precisely in the mode of ac-
tion. He has followed out a train of reasoning by logical
and technical rules, but the signs he has employed are not
pictures of nature, or have lost their original meaning as
such to his mind : he has not seen, as it were, the pro-
cess of nature passing under his eye in an instant of time,
and presented as a whole to his imagination. A familiar
parallel, or an illustration drawn from some artificial or
natural process, of which he has that direct and individual
impression which gives it a reality and associates it with
a name, will, in almost every such case, supply in a mo-
ment this deficient feature, will convert all his symbols
into real pictures, and infuse an animated meaning into
what was before a lifeless succession of words and signs.
We cannot, indeed, always promise ourselves to attain
this degree of vividness in our illustrations, nor are the
points to be elucidated themselves always capable of be-
ing so paraphrased (if we may use the expression) by
any single instance adducible in the ordinary course of
experience ; but the object will at least be kept in view ;
and, as we are very conscious of having, in making such
attempts, gained for ourselves much clearer views of seve-
ral of the more concealed effects of planetary perturba-
tion than we had acquired by their mathematical investi-
gation in detail, we may reasonably hope that the endeavour
will not always be unattended with a similar success in
others.

(10.) From what has been said, it will be evident that
our aim is not to offer to the public a technical treatise,
in which the student of practical or theoretical astronomy
shall find consigned the minute description of methods
of observation, or the formulaj he requires prepared to
his hand, or their demonstrations drawn out in detail. In
all these the present work will be found meagre, and quite
inadequate to his wants. Its aim is entirely different ; be-
ing to present in each case the mere ultimate rationale of
facts, arguments, and processes ; and, in all cases of mathe-

B

14 A TREATISE 0\ ASTRONOMY. [cHAP. I.

muticiil fvpi)liculion, avoidiiiir wliatcver would tend to en-
cunihor its pages with aloebraic or geometrical symbols,
to place under his inspection that central thread of com-
mon sense on which the pearls of analytical research are
invariably strung ; but which, by the attention the latter
claim for themselves, is often concealed from the eye of
the gazer, and not always disposed in the straightest and
most convenient form to follow by those who string thejn.
This is no fault of those who have conducted the inqui-
ries to which we allude. The contention of mind for
which they call is enormous ; and it may, perhaps, be
owing to their experience of hoiv little can be accomplish-
ed in carrying such pi'ocesses on to their conclusion, by
mere ordinary clearness of head; and how necessary it
often is to pay more attention to the purely mathematical
conditions which insure success, — the hooks-and-eyes of
their equations and series, — than to those which enchain
causes with their effects, and both with the human rea-
son,— that Ave must attribute something of that indistinct-
ness of view which is often complained of as a grievance
by the earnest student, and still more commonly ascribed
ironically to the native cloudiness of an atmosphere too
sublime for vulgar comprehension. We think we shall
render good service to both classes of readers, by dissi-
pating, so far as our power lies, that accidental obscurity,
and by showing ordinary untutored comprehension clearly
what it can, and what it cannot, hope to attain.

CHAPTER I.

General Notions — Form and Magnitude of the Earth — Horizon and its
Dip — The Atmosphere — Refraction — Twihght — Appearances result,
ing from diurnal Notion — Parallax — First Step towards forming an
Idea of the Distance of the Stars — Definitions.

(11.) The magnitudes, distances, arrangement, and
motions of the great bodies which make up the visible
universe, their constitution and physical condition, so
far as they can be known to us, with their mutual in-
fluences and actions on each other, so far as they can be

CHAP. I.J GENERAL NOTIONS. 15

traced by the effects produced, and established by legi-
timate reasoning, form the assemblage of objects to
which the attention of the astronomer is directed. The
term astronomy* itself, which denotes the law or rule
of the astra (by which the ancients understood not only
the stars properly so called, but the sun, the moon, and
all the visible constituents of the heavens), sufficiently
indicates this ; and, although the term astrology, which
denotes the reason, theory, or interpretation of the
stars,t has become degraded in its application, and con-
fined to superstitious and delusive attempts to divine
future events by their dependence on pretended plane-
tary influences, tlie same meaning originally attached
itself to that epithet.

(12.) But, besides the stars and other celestial bodies,
the earth itself, regarded as an individual body, is one
principal object of the astronomer's consideration, and,
indeed, the chief of all. It derives its importance, in a
practical as well as theoretical sense, not only from its
proximity, and its relation to us as animated beings,
who draw from it the supply of all our wants, but as the
station from which we see all the rest, and as the only
one among them to which we can, in the first instance,
refer for any determinate marks and measures by which
to recognise their changes of situation, or with which to
compare their distances.

(13.) To the reader who now for the first time takes
up a book on astronomy, it will no doubt seem strange
to class the earth with tlie heavenly bodies, and to as-
sume any community of nature among things apparently
so different. For what, in fact, can be more apparently
different than the vast and seemingly immeasurable ex-
tent of the earth, and the stars, which appear but as
points, and seem to have no size at all ? The earth is
dark and opaque, while the celestial bodies are brilliant.
We perceive in it no motion, while in them we observe
a continual change of place, as we view them at different

* A<rT>if , a Star ; i'ii/«ot, a law ; or i-i^wsn', to tend, as a shepherd his flock ;
so that y-o-Tfovoiioj means "shepherd of tlie stare." The two etymologies
are, however, coincident.

t A o^c,-, reason, or a luorJ, die vehicle of reason; the interpreter of
thtiught.

16 A TREATISE ON ASTRONOMY. [cHAP. I.

hours of the day or night, or at difTercnt seasons of the
year. The ancients, accordingly, one or two of the more
enlightened of them only excepted, admitted no such
community of nature ; and, by thus placing the heavenly
bodies and their movements without the pale of analogy
and experience, effectually intercepted the progress of
all reasoning from what passes here below, to what is
going on in the regions where they exist and move.
Under such conventions, astronomy, as a science of
cause and effect, could not exist, but must be limited to
a mere registry of appearances, unconnected with any
attempt to account for them on reasonable principles.
To get rid of this prejudice, therefore, is the first step
towards acquiring a knowledge of what is really the
case ; and the student has made his first efibrt towards
the acquisition of sound knowledge, when he has learnt
to familiarize himself with the idea that the earth, after
all, may be nothing but a great star. How correct such
an idea may be, and with what limitations and modifica-
tions it is to be admitted, we shall see presently.

(14.) It is evident, that, to form any just notions of
the arrangement, in space, of a number of objects which
we cannot approach and examine, but of which all the
information Ave can gain is by sitting still and watching
their evolutions, it must be very important for us to
know, in the first instance, whether what we call sitting
still is really such : whether the station from which we
view them, with ourselves, and all objects which im-
mediately surround us, be not itself in motion, unper-
ceived by us; and if so, of what nature that motion is.
The apparent places of a number of objects, and their
apparent arrangement with respect to each other, will
of course be materially dependent on the situation of the
spectator among them ; and if this situation be liable to
change, unknown to the spectator himself, an appearance
of change in the respective situations of the objects will
arise, without the reality. If, then, such be actually
the case, it will follow that all the movements we think
we perceive among the stars will not be real movements,
but that some part, at least, of whatever changes of re-
lative place we perceive among them must be merely

CHAP. I.] GENKRAL NOTIONS. 17

apparent, the results of the shifting of our own point of
view ; and that, if we woukl ever arrive at a knowledge
of their real motions, it can only be by first investigating
our own, and making due allowance for its effects.
Thus, the question whether the earth is in motion or at
rest, and if in motion, Avhat that motion is, is no idle in-
quiry, but one on which depends our only chance of
arriving at true conclusions respecting the constitution
of the universe.

(15.) Nor let it be thought strange that we should
speak of a motion existing in the earth, unperceived by
its inhaljitants : we must remember that it is of the earth
us a ivhole, with all that it holds within its substance,
or sustains on its surface, that we are speaking ; of a
motion common to the solid mass beneath, to the ocean
which flows around it, the air that rests upon it, and the
clouds which float above it in the air. Such a motion,
which should displace no terrestrial object from its re-
lative place among others, interfere with no natural pro-
cesses, and produce no sensations of shocks or jerks,
might, it is very evident, subsist undetected by us.
There is no peculiar sensation which advertises us that
we are in motion. We perceiv6 jerks, or shocks, it is
true, because these are sudden changes of motion, pro-
duced, as the laws of mechanics teach us, by sudden
and powerful forces acting during short times ; and these
forces, applied to our bodies, are what wefecl. When,
for example, we are carried along in a carriage with the
blinds down, or with our eyes closed (to keep us from
seeing external objects), we perceive a tremor arising
from inequalities in the road, over which the carriage is
successively lifted and let fall, but we have no sense of
progress. As the road is smoother, our sense of motion
is diminished, though our rate of travelling is accelerated.
Those who have travelled on the celebrated rail-road
between Manchester and Liverpool testify that but for
the noise of the train, and the rapidity with which ex-
ternal objects seem to dart by them, the sensation is al-
most that of perfect rest.

(16.) But it is on shipboard, where a great system is
maintained in motion, and where we are surrounded

b2

18 A TREATISE ON ASTRONOMY. [CHAP. I.

with a multitude of objects wliich participate with our-
selves and each other in the common progress of the
whole mass, that we feel most satisfactorily the identity
of sensation between a state of motion and one of rest.
In the cabin of a large and heavy vessel, going smoothly
before the wind in still water, or drawn along a canal,
not the smallest indication acquaints us with the way it
is making. We read, sit, walk, and perform every cus-
tomary action as if we were on land. If we throw a
ball into the air, it falls back into our hand ; or, if avc
drop it, it lights at our feet. Insects buzz around us as
in the free air ; and smoke ascends in the same manner
as it would do in an apartment on shore. If, indeed,
we come on deck, the case is, in some respects, difler-
ent ; the air, not being carried along with us, drifts away
smoke and other light bodies — such as feathers aban-
doned to it — apparently, in the opposite direction to that
of the ship's progress ; but, in reality, they remain at
rest, and we leave them behind in the air. Still, the
^.illusion, so far as massiye objects and our own move-
ments are concerned, remains complete ; and when we
look at the shore, we then perceive the effect of our own
motion transferred, in a contrary direction, to external
objects — external, that is, to the system of which ive
form a part.

" Provehimur portu, terraeque urbesque recediuit."

(17.) Not only do external objects at rest appear in
motion generally, with respect to ourselves when we
are in motion among them, but they appear to move one
among the other — they shift their relative apparent
places. Let any one travelling rapidly along a high
road fix his eye steadily on any object, but at the same
time not entirely withdraw his attention from the gene-
ral landscape, — he Avill see, or think he sees, the whole
landscape thrown into rotatioji, and moving round that
object as a centre ; all objects between it and himself
appearing to move backwards, or the contrary way to
his own motion ; and all beyond it, forwards, or in the
direction in which he moves : but let him withdraw his
eye from that object, and fix it on another, — a nearer

CHAP. I.] FORM OF THE EARTH. 19

one, for instance, — immediately the appearance of ro-
tation shifts also, and the apparent centre about which
this illusive circulation is performed is transferred to the
new object, which, for the moment, appears to rest.
This apparent change of situation of objects with re-
spect to one another, arising from a motion of the spec-
tator, is called a parallactic motion ; and it is, therefore,
evident that, before we can ascertain whether external
objects are really in motion or not, or what their mo-
tions are, we must subduct, or allow for, any such pa-
rallactic motion which may exist.

(18.) In order, however, to conceive the earth as in
motion, we must form to ourselves a conception of its
shape and size. Now, an object cannot have shape and
size, unless it is limited on all sides by some definite
outline, so as to admit of our imagining it, at least, dis-
connected from other bodies, and existing insulated in
space. The first rud(? notion we form of the earth is
that of a flat surface, of indefinite extent in all directio;is-"
from the spot where we stand, above which are the air
and sky ; below, to an indefinite profundity, solid mat-
ter. This is a prejudice to be got rid of, like that of the
earth's immobility ; but it is one much easier to rid our-
selves of, inasmuch as it originates only in our own
mental inactivity, in not questioning ourselves u)here we
will place a limit to a thing we have been accustomed
from infancy to regard as immensely large ; and does
not, like that, originate in the testimony of our senses
unduly interpreted. On the contrary, the direct testi-
mony of our senses lies the other way. When we see
the sun set in the evening in the west, and rise again in
the east, as we cannot doubt that it is the same sun we
see after a temporary absence, we must do violence to
all our notions of solid matter, to suppose it to have
made its way through the substance of the earth. It
must, therefore, have gone under it, and that not by a
mere subterraneous channel ; for if we notice the points
where it sets and rises for many successive days, or for
a whole year, we shall find them constantly shifting,
round a very large extent of the horizon ; and, besides,
the moon and stars also set and rise again in all points

20 A TRKATISE ON ASTRONOMY. [cHAP. I.

of the visible horizon. The ronrhision is plain : the
earth cannot extend indefinitely in depth downwards,
nor indefinitely in surface laterally ; it must have not
only bounds in a horizontal direction, but also an under
SiV/f, round which the sun, moon, and stars can pass;
and that side must, at least, be so far like what we see,
that it must have a sky and sunshine, and a day when it
is night to us, and vice versa ; where, in short,

— " redit a nobis Aurora, diemque reducit.
Nosque uhi primus equis orieiis affiavit anhelis,
lUic sera ruberis acceiidit lumina Vesper." Georg.

(19.) As soon as we have familiarized ourselves with
the conception of an earth wilhont foundaf ions or fixed
supports — existing insulated in space from contact of
every thing external, it becomes easy to imagine it in
motion — or, rather, difiicult to imagine it otherwise ; for,
since there is nothing to retain it in one place, should
any causes of motion exist, or any forces act upon it, it
must obey their impulse. Let us next see what obvious
circumstances there are to help us to a knowledge of the
shape of the earth.

(20.) Let us first examine Avhat we can actually see
of its shape. Now, it is not on land (unless, indeed, on
uncommonly level and extensive plains) that Ave can see
any thing of the general figure of the earth ; — the hills,
trees, and other objects which roughen its surface, and
break and elevate the line of the horizon, though ob-
viously bearing a most minute proportion to the u'hole
earth, are yet too considerable, with respect to ourselves
and to that small portion of it which we can see at a sin-
gle view, to allow of our forming any judgment of the
form of the whole, from that of a part so disfigured.
But with the surface of the sea, or any vastly extended
level plain, the case is otherwise. If we sail out of sight
of land, whether we stand on the deck of the ship or
climb the mast, we see the surface of the sea — not losing
itself in distance and mist, but terminated by a sharp,
clear, well defined line, or offing as it is called, Avhich
runs all round us in a circle, having our station for its
centre. That this line is really a circle, we conclude,
first, from the perfect apparent similarity of all its parts ;

CHAP. I.] HORIZON AND ITS DIP. 21

and, secondly, from the fact of all its parts appearing at
the same distance from us, and that evidently a mode-
rate one; and, thirdly, from this, that its apparent
diameler, measured with an instrument called the dijJ
sector, is the same (except under some singular atmo-
spheric circumstances, which produce a temporary distor-
tion of the outline), in whatever direction the measure
is taken, — properties which belong only to the circle
among geometrical figures. If we ascend a high emi-
nence on a plain (for instance, one of the Egyptian py-
ramids), the same holds good.

(21.) Masts of ships, however, and the edifices erected
by man are trifling eminences compared to what nature
itself affords; jElna, Teneriff'e, Mowna Roa, are emi-
nences from which no contemptible aliquot part of the
whole earth's surface can be seen ; but from these again
— in those few and rare occasions when the transparency
of the air will permit the real boundary of the horizon,
the true sea-line, to be seen — the very same appearances
are witnessed, but with this remarkable addition, viz.
that the angular diameter of the visible area, as mea-
sured by the dip sector, is materially less than at a lower
level, or, in other words, that the apparent size of the
earth has sensibly diminished as we have receded from
its surface, while yet the absolute quantity of it seen at
once has been increased.

(22.) The same appearances are observed universally,
in every part of the earth's surface visited by man.
Now, the figure of a body which, however seen, ap-
pears always circular, can be no other than a sphere or
globe.

(23.) A diagram will elucidate this. Suppose the
earth to be represented by the sphere LHNQ, whose
centre is C, and let A, G, M be stations at different
elevations above various points of its surface, represent-
ed by a, £f, m, respectively. From each of them (as
from M) let a line be drawn, as MNn., a tangent to the
surface at N, then will this line represent the visual ray
along which the spectator at M will see the visible ho-
rizon ; and as this tangent sweeps round M, and comes
successively into the positions MOo,MV p,M Q,q, the

22 A TREATISE ON ASTROKOMV. [cHAP. I.

point of contact N will murk out on the surface the circle
N O P Q. The area of this circle is the portion of the

eartli's surface visible to a spectator at M, and the angle
NMQ included between the two extreme visual rays is
the measure of its ap])arent angular diameter. Leaving,
at present, out of consideration the effect of refraction in
the air below M, of which more hereafter, and which
always tends, in some degree, to increase that angle, or
render it more obtuse, this is the angle measured by the
dip sector. Now, it is evident, 1st, that as the point M
is more elevated above m, the point immediately below
it on the sphere, the visible area, i. e. the spherical seg-
ment or slice NOPQ, increases ; 2dly, that the distance
of the visible horizon* or boundary of our view from
the eye, viz. the line MN, increases ; and, 3dly, that
the angle MNQ becomes less obtuse, or, in other words,
the apparent angular diameter of the earth diminishes,

'Ogi^ji, to terminate

CHAP. 1,J HORIZON AND ITS DIP. 23

being nowhere so great as 180°, or two right angles,
but falling short of it by some sensible quantity, and that
more and more the hio'her we ascend. The figure ex-
hibits three states or stages of elevation, with the hori-
zon, &c. corresponding to each, a glance at which will
explain our meaning ; or, limiting ourselves to the larger
and more distinct, MNOPQ, let the reader imagine
?iNM, MQq to be the two legs of a ruler joined at M,
and kept extended by the globe NmQ, between them.
It is clear, that as the joint M is urged home towards
the surface, the legs will open, and the ruler will become
more nearly straight, but will not attain perfect straight-
ness till M is In-ought fairly up to contact with the sur-
face at m, in which case its whole length will become a
tangent to the sphere at m, as is the line xy.

(24.) This explains what is meant by the dip of the
horizon. M?», which is perpendicular to the general
surface of the sphere at m, is also the direction in which
a plumb-line* would hang ; for it is an observed fact,
that in all situations, in every part of the earth, the di-
rection of a plumb-line is exactly perpendicular to the
surface of still-water; and moreover, that it is also ex-
actly perpendicular to a line or surface truly adjusted by
a spirit-level. * Suppose, then, that at our station M we
were to adjust a line (a wooden ruler for instance) by a
spirit-level, with perfect exactness ; tlien, if we suppose
the direction of this line indefinitely prolonged both
ways, as XMY, the line so drawn will be at right
angles to Mm, and therefore parallel to xmy, the tan-
gent to the sphere at m. A spectator placed at M will
therefore see not only all the vault of the sky above this
line, as XZY, but also that portion or zone of it which
lies between XN and YQ ; in other words, his sky will
be more than a hemisphere by the zone YQXN. It is
the angular breadth of this redundant zone — the angle
YMQ, by which the visible horizon appears depressed
below the direction of a spirit-level — that is called the
dip of the horizon. It is a correction of constant use in
nautical astronomy. •

* See this instrument described in chap. II.

24 A TREATISE ON ASTRONOMY. [ciIAP. 1.

(25.) From the foregoing explanations it appears,
1st, That the general figure of the earth (so far as it can
be gathered from this kind of observation) is that of a
sphere or globe. In this we also include that of the sea,
which, wherever it extends, covers and fills in those in-
equalities and local irregularities which exist on land,
but which can of course only be regarded as trifling de-
viations from the general outline of the whole mass, as
we consider an orange not tlie less round for tlie rough-
nesses on its rind. 2dly, That the appearance of a visi-
ble horizon, or sea ofling, is a consequence of the cur-
vature of the surface, and does not arise from the inability
of the eye to follow objects to a greater distance, or
from atmospheric indistinctness. It will be worth v.diile
to pursue the general notion thus acquired into some of
its consequences, by which its consistency with obser-
vations of a different kind, and on a larger scale, will be
put to the test, and a clear conception be formed of the
manner in which the parts of the earth are related to
each other, and held together as a whole.

(26.) In the first place, then, every one who has passed
a little while at the sea side is aware that objects may be
seen perfectly well beyond the offing or visible horizon
— but not the ivhole of them. We only see their upper
parts. Their bases where they rest on, or rise out of
the water, are hid from view by the spherical surface of
the sea, which protrudes between them and ourselves.
Suppose a ship, for instance, to sail directly away from
our station ; — at first, when the distance of the ship is
small, a spectator, S, situated at some certain height
above the sea, sees the whole of the ship, even to the
water line where it rests on the sea, as at A. As it re-
cedes it diminishes, it is true, in apparent size, but still
the zvhole is seen down to the water line, till it reaches
the visible horizon at B. But as soon as it has passed
this distance, not only does the visible portion stiil con-
tinue to diminish in apparent size, but tlie hull begins to
disappear bodily, as if sunk below the surface. When
it has reached a certain distance, as at C, its hull
has entirely vanished, but the masts and sails remain,
presenting the appearance c. But if, in this state of

CHAP,

^•J

SIZE OF THE EARTH.

25

things, the spectator quickly ascends to a higher station,
T, whose visible horizon is at D, the hull comes again
in sight; and when he descends again he loses it. The
ship still receding, the lower sails seem to sink below
the water, as at d, and at length the whole disappears :
Avhile yet the distinctness with which the last portion of
the sail d is seen is such as to satisfy us that were it not
for the interposed segment of the sea, ABODE, the dis-
tance TE is not so great as to have prevented an equally
perfect view of the whole.

(27.) In this manner, therefore, if we could measure
the heights and exact distance of two stations which
could barely be discerned from each other over the edge
of the horizon, we could ascertain the actual size of the
earth itself: and, in fact, were it not for the effect of re-
fraction, by which we are enabled to see in some small
degree round the interposed segment fas will, be here-
after explained), this would be a tolerably good method
of ascertaining it. Suppose A and B to be two emi-

nences, whose perpendicular heights Art and B h (which,
for simplicity, we will suppose to be exactly equal) are
knov/n, as well as their exact horizontal interval aD6,

C

aO A TREATISE ON ASTRONOMY. [cHAP. I,

by measurement; then is it. clear that D, the visi-
ble horizon of both, will lie just half-way between
them, and if we suppose aDb to be the sphere of
the earth, and C its centre in the figure CUbB, we
know Db. the length of the arch of the circle between
D and b, — viz. half the measured interval, and bB, tho
excess of its secant above its radius — which is the height
of B, — data which, by tlie solution of an easy geometrical
problem, enable us to find the length of the radius DC.
If, as is really the case, we suppose both the heights and
distance of the stations inconsiderable in comparison with
the size of the earth, the solution alluded to is contained
in the following proposition : —

The eurtlt's diameter bears the same proportion to the
distance of the visible horizon from the eye as that dis-
tance does to the height of the eye above the sea level.

When the stations are unequal in height the problem
is a little more complicated.

(28.) Although, as we have observed, the effect of
refraction prevents this from being an exact method of
ascertaining the dimensions of the eartli, yet it will suf-
fice to aflbrd such an approximation to it as shall be of
use in the present stage of the reader's knowledge, and
help him to many just conceptions, on which account
we shall exemplify its application in numbers. Now, it
appears by observation, that two points, each ten feet
above the surface, cease to be visible from each other
over still water, and in average atmospheric circum-
stances, at a distance of about 8 miles. But 10 feet is
Jhe 528th part of a mile, so tliat half their distance, or
4 miles, is to the height of each as 4 x 528 or 2112 : 1,
and therefore in the same proportion to 4 miles is the
length of the earth's diameter. It must, therefore, be
equal to 4 X 2112 = 8448, or, in round numbers, about
8000 miles, which is not very far from the truth.

(29.) Such is the first rough result of an attempt to
ascertain the earth's magnitude ; and it will not be amiss
if we take advantage of it to compare it with objects we
have been accustomed to consider as of vast size, so as
to interpose a few steps between it and our ordinary ideas

CHAP. l.J VISIBLE PORTION OF THE SURFACE. 27

of dimension. We have before likened the inequalities
on tlie earth's surface, arising from mountains, valleys,
buildings, &c. to the roughnesses on the rind of an
orange .''compared with its general mass. The compa-
rison is quite free from exaggeration. The highest moun-
tain known does not excectf five miles in perpendicular
elevation: this is only one 1600th part of the earth's
diameter ; consequently, on a globe of sixteen inches in
' diameter, such a mountain would be represented by a
protuberance of no more than one hundredth part of an
inch, which is about the thickness of ordinary drawing-
paper. Now as there is no entire continent, or even any
very extensive tract of land, known, whose general ele-
vation above the sea is any thing like half this quantity,
it follows, that if we would construct a correct model of
our earth, with its seas, continents, and mountains, on
a globe sixteen inches in diameter, the whole of the land,
with the exception of a few prominent points and ridges,
must be comprised on it within the thickness of thin
writing paper; and the highest hills would be represented
by tlie smallest visible grains of sand.

(30.) The deepest mine existing does not penetrate
half a mile below the surface : a scratch, or pin-hole,
duly representing it, on the surface of such a globe as
our model, would be impercepti])le without a magnifier.
(.31.) The greatest depth of sea, probably, does not
much exceed the greatest elevation of the continents ;
and would, of course, 1)e represented by an excavation,
in about the same proportion, into the substance of the
globe : so that the ocean comes to be conceived as 'a
mere film of liquid, such as, on our model, would be leu
by a brush dipped in colour and drawn over those parts
intended to represent the sea : only in so conceiving it,
we must bear in mind that the resemblance extends no
farther than to proportion in point of quantity. The
mechanical laws which would regulate the distribution
and movements of such a film, and its adhesion to the
surface, are altogether difl'erent from those which govern
me phenomena of the sea.

(32.) Lastly, the greatest extent of the earth's surface
which has ever been seen at once by man, was that ex-

28 A TREATISE ON ASTRONOMY. [cHAP. 1.

posed to the view of MM. Biot and Gay-Lussac, in tlieir
celebrated aeronautic expedition to the enoi'mous heiglit
of 25,000 feet, or rather less than five miles. To esti-
mate the proportion of tlie area visible from this elevation
to the whole earth's surface, we must have recourse to
the geometry of the sphere, which informs us that the
convex surface of a spherical segment is to the whole sur-
face of the sphere to which it belongs as the versed sine
or thickness of the segment is to the diameter of the
sphere ; and further, that this thickness, in the case we
are considering, is almost exactly equal to the perpen-
dicular elevation of the point of sight alwve the surface.
The proportion, therefore, of the visilile area, in this
case, to the whole earth's surface, is that of five miles to
8000, or 1 to 1600. Tlie portion visible from ^tna, the
Peak of Teneriffe, or Mowna Roa, is about one 4000th.
(33.) When we ascend to any very considerable ele-
vation above the surface of the earth, either in a balloon,
or on mountains, we are made aware, by many uneasy
sensations, of an insufliicient supply of ah\ The barome-
ter, an instrument which informs us of the weight of air
incumbent on a given horizontal surface, confirms this im-
pression, and affords a direct measure of the rate of dimi-
nution of the quantity of air which a given space includes
as we recede from the surface. From its indications we
learn, that when we have ascended to the height of 1000
feet, we have left below us about one thirtietli of the
whole mass of the atmosphere : — that at 10,600 feet of
perpendicular elevation (which is rather less than that of
the summit of iEtna*) we have ascended through about
one third ; and at 18,000 feet (which is nearly that of Co-
topaxi) through one half the material, or, at least, ihe
ponderable, body of air incumbent on the earth's surface.
Fi'om the progression of these numbers, as well as, a pri-
ori, from the nature of the air itself, which is compressi-
ble, i. e. capable of being condensed, or crowded into a
smaller space in proportion to the incumbent pressure, it
is easy to see that, although by rising still higher we should

* Tlie height of /Etna above the Mediterriinean (as it results from a
barometrical measurement of my own, made in July, 1824, under very
favourable circumstances) is 10,872 English feet. — Author.

CHAP. I.J THE ATMOSPHERK. 29

continually get above more and more of the air, and so re-
lieve ourselves more and more from the pressure Avith
wliich it weighs upon us, yet the amount of this additional
relief, or \he ponderable quantity o{ 2\x surmounted, would
be by no means in proportion to the additional height as-
cended, but in a constantly decreasing ratio. An easy
calculation, however, founded on our experimental know-
ledge of the properties of air, and the mechanical laws
whicli regulate its dilation and compression, is sufficient
to sliow that, at an altitude above the surface of the earth
not exceeding the hundreth part of its diameter, the tenui-
ty, or rarefiction, of the air must be so excessive, that not
only animal life could not subsist, or combustion be main-
tained in it, but that the most delicate means we possess of
ascertaining the existence of any air at all would fail to
allurd the slightest perceptible indications of its presence.

(34.) Laying out of consideration, therefore, at pre-
sent, all nice questions as to the probable existence of a
definite limit to the atmosplicre, beyond which tliere is,
absolutely and rigorously speaking, no air, it is clear, that,
for all practical purposes, we may speak of those regions
which are more distant above the earth's surface than the
hundredth part of its diameter as void of air, and of course
of clouds (which are nothing but visible vapours, diffused
and floating in tlie air, sustained by it, and rendering
it iurljid as mud does water). It seems probable, from
many indications, that the greatest height at which visible
clouds ever exist does not exceed ten miles ; at which
height the density of tlie air is about an eighth part of
what it is at the level of the sea.

(.S5.) We are thus led to regard the atmosphere of air,
with the clouds it supports, as constituting a coating of
equable or nearly equal)le thickness, enveloping our globe
on all sides ; or rat)u3r as an aerial ocean, of which the
surface of the sea and laud constitutes the bed, and whose
inferior portions or strata, within a few miles of the
earth, contain by far tlie greater part of the Avhole mass,
the density diminishing Avith extreme rapidity as we re-
cede upwards, till, witliin a very moderate distance (such
as would be represented I))^ the sixth of an inch on the mo-
del we have before spoken of, and which is not more in pro-

c 3

30 A TREATISE ON ASTRONOMY. [cHAP. I.

portion to the globe on which it rests, than the downy
skin of a peach in comparison Avitli the fruit within it),
all sensible trace of the existence of air disappears.

(.36.) Arguments, however, are not wanting to render
it, if not absolutely certain, at least in the highest degree
probable, that the surface of the aerial, like that of the
aqueous ocean, has a real and definite limit, as above hint-
ed at ; beyond which there is positively no air, and above
which a fresh quantity of air, could it be added from with-
out, or carried aloft from below, instead of dilating itself
indefinitely upwards, would, after a certain very enor-
mous but still finite enlargement of volume, sink and
merge, as water poured into the sea, and distribute itself
among the mass beneath. With the truth of this conclu-
sion, however, astronomy has little concern ; all tlie ef-
fects of the atmosphere in modifying astronomical phe-
nomena being the same, whether it be supposed of defi-
nite extent or not.

(37.) Moreover, whichever idea we adopt, it is equally
certain that, within those limits in which it possesses any
appreciable density, its constitution is the same over all
points on the earth's surface ; that is to say, on the great
scale, and leaving out of consideration temporary and local
causes of derangement, such as winds, and great fluc-
tuations, of the nature of waves, which prevail in it to an
immense extent : in other words, that the law of diminu-
tion of the air's density as we recede upwards from the
level of the sea is the same in every column into which
we may conceive it divided, or from whatever point of
the surface we may set out. It may therefore be consi-
dered as consisting of successively superposed strata or
layers, each of the form of a spherical shell, concentric
with the sfeneral surface of the sea and land, and each of
which is rarer, or specifically lighter, than that immedi-
ately beneath it ; and denser, or specifically heavier, than
that immediately above it. This kind of distribution of
its ponderal)le mass is necessitated by the laws of the
equilibrium of fluids, whose results barometric observa-
tions demonstrate to be in perfect accordance with expe-
rience.

It must be observed, however, that with this distribu-

CHAP.

^•]

REFRACTION.

31

lion of its strata tlie inequalities of mountains and valleys
have no concern ; these exercise no more influence in
modifying their general spherical figure than the inequali-
ties at the bottom of the sea interfere with the general
sphericity of its surface.

(38.) It is the power which air possesses, in common
witli all transparent media, of refracting the rays of light,
or bending them out of their straight course, which renders
a knowledge of the constitution of the atmosphere import-
ant to the astronomer. Owing to this property, objects
seen obliquely through it appear otherwise situated than
they would to the same spectator, had the atmosphere no
existence ; it thus produces a false impression respecting
tlieir places, v/hich must be rectified by ascertaining the
amount and direction of the displacement so apparently
produced on each, before we can come at a knowledge
of the true directions in which they are situated from us
at any assigned moment.

(39.) Suppose a spectator placed at A, any point of the
earth's surface KAA;, and let hi. Mm, N?*, represent

the successive strata or layers, of decreasing density, into
which we may conceive the atmosphere to be divided,
and which are spherical surfaces concentric with KA", the
earth's surface. Let S represent a star, or other heavenly
body, beyond the utmost limit of the atmosphere ; then,
if the air were away, the spectator would see it in the di-

32 RKFRACTION. [cHAP. I,

rection of the straight line AS. But, in reality, when
the ray of light SA reaches tlie atmosplicrc, suppose at d,
it will, hy the laws of optics, begin to bend downwards,
and take a more inclined (Urection, as d c. .This bending
will at first be imperceptible, owing to the extreme tenu-
ity of the uppermost strata ; but as it advances downwards,
the strata continually increasing in density, it will continu-
ally undergo greater and greater refraction in the same di-
rection ; and thus, instead of pursuing the straight line
S(/A, it will describe a curve Sdcb a, continually more
and more concave downwards, and will reach the earth,
not at A, but at a certain point a, nearer to S, This ray,
consequently, will not reach the spectator's eye. Tlie ray
by which he will see the star is, therefore, not Sc/A, but
another ray which, had there been no atmosphere would
have struck the earth at K, a point ^e/imrZ the spectator ;
but which, being bent by the air into the curve SDCBA,
actually strikes on A. Now, it is a law of optics, that an
object is seen in the direction which the visual ray has at
the instant of arriving at the eye, without regard to what
may have I^ecn otherwise its course between the object and
the eye. Hence the star S will be seen, not in the di-
rection AS, but in that of As, n tangent to the curve
SDCBA, at A. But because the curve described by the
refracted ray is concave downwards, the tangent As, will
lie above AS, the unrefracted ray : consequently the object
S will appear more elevated above the horizon AH, when
seen through the refracting atmosphere, dian it would ap-
pear were there no such atmosphere. Since, however, the
disposition of the strata is the same in all directions around
A, the visual ray will not be made to deviate laterally, but
will remain constantly in the same vertical plane SAC,
passing through the eye, the object, and the earth's centre.
(40.) The effect of the air's refraction, then, is to raise
all the heavenly bodies higher above the horizon in ap'
pearance than they are in reality. Any such body, situ-
ated actually in the true horizon, will appear above it, or
will have some certain apparent altitude (as it is called).
Nay, even some of those actually below the horizon, and
which woidd therefore be invisible but for the effect of
refraction, are, by that efl'ect, raised altove it and brought

CHAP. I.J REFRACTION. 33

into sight. Thus, the sun, when situated at P below the
true horizon, All, of the spectator, becomes visible to him,
as if it stood at p, by the refracted ray FqrtA, to which
Ap is a tangent.

(41.) The exact estimation of the amount of atmo-
spheric refraction, or the strict determination of the angle
SAs, by which a celestial object at any assigned altitude,
HAS, is raised in appearance above its true place, is, un-
fortunately, a very difficult subject of physical inquiry,
and one on which geometers (from Avhom alone we can
look for any information on the subject) are not yet en-
tirely agreed. The difficulty arises from this, that the
density of any stratum of air (on which its refracting
power depends) is affected not merely by the superincum-
bent pressure, but also by its temperature or degree of
heat. Now, although we know that as we recede from the
earth's surface the temperature of the air is constantly
diminishing, yet the law, or amount of this diminution
at different heights, is not yet fully ascertained. More-
over, the refracting power of air is perceptibly affected by
its moisture ; and this, too, is not the same in every part of
an aerial column ; neither are we acquainted with the laws
of its distribution. The consequence of our ignorance on
these points is to introduce a corresponding degree of
uncertainty into the determination of the amount of refrac-
tion which affects, to a certain appreciable extent, our
knowledge of several of the most important data of as-
tronomy. The uncertainty thus induced is, however,
confined within such very narrow limits as to be no cause
of embarassment, except in the most delicate inquiries,
and to call for no further allusion in a treatise like the
present.

(42.) A "Table of Refractions," as it is called, or a
statement of the amount of apparent displacement aris-
ing from this cause, at all altitudes, or in every situation
of a heavenly body, from the horizon to the zenith,'^ or
point of the sky vertically above the spectator, and, under
all the circumstances in which astronomical observations
are usually performed which may influence the result, is

* From an Arabic word of this signification.

34 A TREATISE ON ASTRONOMY. [cHAP. I.

one of the most important and indispensable of all astro-
nomical tables, since it is only by the nse of such a table
we are enabled to get rid of an illusion which must
otherwise pervert all our notions respecting the celestial
motions. Such have been, accordingly, constructed with
great care, and are to be found in every collection of
astronomical tables.* Our design, in the present treatise,
will not admit of the introduction of tables ; and we
must, therefore, content ourselves here, and in similar
cases, with referring the reader to works especially des-
tined to furnish these useful aids to calculation. It is,
however, desirable that he should bear in mind the
following general notions of its amount, and law of
variation.

(43.) 1st. In the zenith there is no refraction; a ce-
lestial object, situated vertically over head, is seen in its
true direction, as if there were no atmosphere.

2dly. In descending from the zenith to the horizon,
the refraction continually increases ; objects near the
horizon appearing more elevated by it above their true
directions than those at a high altitude.

3dly. The rate of its increase is nearly in proportion
to the tangent of the apparent angular distance of the
object from the zenitli. But this rule, which is not far
from tlie tiiith, at moderate zenith distances, ceases to
give correct results in the vicinity of the horizon, where
the law becomes much more complicated in its ex-
pression.

4thly. The average amount of refraction, for an ob-
ject half-wa)^ between the zenith and liorizon, or at an
apparent altitude of 45°, is about 1' (more exactly 57"),
a quantity hardly sensible to the naked eye ; but at the
visible horizon it amounts to no less a quantity than 33',
which is rather more than the greatest apparent diameter
of either the sun or the moon. Hence it follows, that
when we see the lower edge of the sun or moon just ap-
parently resting on tlie horizon, its whole disk is in
reality below it, and would be entirely out of sight and

*Vide " Requisite Tables to be used with the Nautical Almanac."
See a]po Nautical Almanac for 1833, Dr. Pearson's Astronomical Tables,
and Mr. Baily's Astronomical Tables and FoitouIbd,

CHAP. I.] TWILIGHT. 35

concealed by the convexity of the earth but for the bend-
ing round it, which the rays of light have undergone in
neir passage through the air, as alluded to in art. 40.

(44.) It follows from this, that one obvious effect of
refraction must be to shorten the duration of night and
darkness, by actually prolonging the stay of the suii and
moon above the horizon. But even after they are set,
the inlluence of the atmosphere still continues to send
us a poflion of their light; not, indeed, by direct trans-
mission, but by reflection upon the vapours, and minute
solid particles, which float in it, and, perhaps, also on
the actual material atoms of the air itself. To understand
how this takes place, we must recollect, that it is not
only by the direct light of a luminous object that we
see, but that whatever portion of its light which would
not otherwise reach our eyes, is intercepted in its course,
and thrown back, or laterally, upon us, becomes to us a
means of illumination. Such reflective obstacles always
exist floating in the air. Tlie whole course of a sun-
beam penetrating through the chink of a window-shutter
into a dark room, is visible as a bright line in the air;
and even if it be stifled, or let out through an opposite
crevice, the liglit scattered through the apartment from
this source is sufficient to prevent entire darkness in the
room. The luminous lines occasionally seen in the air,
in a sky full of partially broken clouds, which the vulgar
term " the sun drawing water," are similarly caused.
They are sunbeams, through apertures in clouds, par-
tially intercepted and reflected on the dust and vapours
of the air below. Thus it is'with those solar rays which,
after the sun is itself concealed by the convexity of the
earth, continue to traverse the higher regions of the at-
mosphere above our heads, and pass through and out of
it, without directly striking on the earth at all. Some
portion of them is intercepted, and reflected by the float-
ing particles above mentioned, and thrown back, or la-
terally, so as to reach us, and afford us that secondary
illumination, which is twilight. The course of such rays
will be immediately understood from the annexed figure,
in which ABCD is the earth ; A a point on its surface,
where the sun S is in the act of setting ; its last lower

36 A TREATISE ON ASTRONOMY. [cHAP. I

ray SAM just grazing the surface at A, while its superior
rays SN, SO, traverse the atmosphere above A withoui
striking the earth, leaving it finally at the points PQR,
after being more or less bent in passing through it, the

lower most, the higher less, and that which, like SRO,
merely grazes the exterior limit of the atmosphere, not
at all. Let us consider several points, A, B, O, D, each
more remote than the last from A, and each more deeply
involved in the earth's shadoiv, which occupies the whole
space from A beneath the line AM. Now, A just receives
the sun's last direct ray, and, besides, is illuminated by
the whole reflective atmosphere PQRT. It therefore
receives tvv^ilight from the whole sky. The point B, to
which the sun has set, receives no direct solar light, nor
any, direct or reflected, from all that part of its visible
atmosphere which is below APM ; but from the lenti-
cular portion PR.r, Avhich is traversed by the sun's rays,
and which lies above the visible horizon BR of B, it re-
ceives a twilight, which is strongest at R, the point im-
mediately below which the sun is, and fades away gradu-
ally towards P, as the luminous part of the atmosphere
thins off". At C, only the last or thinnest portion, PQ.?
of the lenticular segment, thus illuminated, lies above
the horizon, CQ, of that place : here, then, the twilight
is feeble, and confined to a small space in and near the

CHAP. 1.] TERRESTRIAL REFRACTIOX. St

horizon, which the sun has quitted, wliile at D the twi*
light has ceased altogether.

(45.) When the sun is above the horizon, it illumi*
nates the atmosphere and clouds, and these again dis-
perse and scatter a portion of its light in all directions,
so as to send some of its rays to every exposed pointy
from every point of the sky. The generally diflused
light, therefore, which we enjoy in the daytime, is a phe-
nomenon originating in the very same causes as the twi-
light. Were it not for the reflective and scattering power
of the atmosphere, no objects would be visible to us out
of direct sunshine ; every shadow of a passing cloud
would be pitchy darkness ; the stars Avould be visible all
day, and every apartment, into which the sun had not di-
rect admission, would be involved in nocturnal obscurity.
This scattering action of the atmosphere on the solar
light, it should be observed, is greatly increased by the
irregularity of temperature caused by the same luminary
in its different parts, which, during the daytime, throws
it into a constant state of undulation, and, bv thus brinof-
ing together masses of air of very unequal temperatures,
produces partial reflections and refractions at their com-
mon boundaries, by which much light is turned aside
from the direct course, and diverted to the purposes of
general illumination.

(46.) From the explanation we have given, in arts*
39 and 40, of the nature of atmospheric refraction,
and the mode in which it is produced in the progress
of a ray of light through successive strata, or layers of
the atmosphere, it will be evident, that whenever a ray
passes obliquely from a higher level to a lower one, or
vice versa, its course is not rectilinear, but concave
downwards ; and of course any object seen by means of
such a ray, must appear deviated from its true place,
whether that object be, like the celestial bodies, entirely
beyond the atmosphere, or, like the summits of moun-
tains, seen from the plains, or other terrestrial stations,
at different levels, seen from each other, immersed in it.
Every difference of level, accompanied, as it must be,
with a difference of density in the aerial strata, must also
have, corresponding to it, a certain amount of refraction i

D

38 A TKEATI3E ON ASTROXCtlY. [cHAP. I-

less, indeed, than what would he produced hy the whole
atmosphere, but still often of very appreciable, and even
considerable, amount. This refraction between terres-
trial stations is termed ferresfrial refraction, to distin-
guish it from that total effect which is only produced on
celestial objects, or such as are beyond the atmosphere,
and which is called celestial or astronomical refraction.

(47.) Another effect of refraction is to distort the visi-
ble forms and proportions of objects seen near the hori-
zon. The sun, for instance, which, at a considerable
altitude, always appears round, assumes, as it approaches
the horizon, a flattened or oval outline ; its horizontal
diameter being visibly greater than that in a vertical di-
rection. When very near tlie horizon, this flattening is
evidently more considerable on the lower side than on
the upper ; so that the apparent form is neither circular
nor elliptic, but a species of oval, which deviates more
from a circle below than above. This singular effect,
which any one may notice in a fine sunset, arises from the
rapid rate at which the refraction increases in approach-
ing the horizon. Were every visible point in the sun's
circumference equally raised by refraction, it would still
appear cii'cular, though displaced : but the lower portions
being more raised than the upper, the vertical diameter is
thereby shortened, while the two extremities of its hori-
zontal diameter are equally raised, and in parallel direc-
tions, so that its apparent length remains the same. The
dilated size (generally) of the sun or moon, when seen
near the horizon, beyond what they appear to have
when high up in the sky, has nothing to do with refrac-
tion. It is an illusion of the judgment arising from the
terrestrial objects interposed, or placed in close compari-
son with them. In that situation we view and judge of
them as we do of terrestrial objects — in detail, and with
an acquired habit of attention to parts. Aloft we have
no associations to guide us, and their insulation in the
expanse of sky leads us rather to undervalue than to
overrate their apparent magnitudes. Actual measure-
ment with a proper instrument corrects our error, with-
out, however, dispelling our illusion. By this we learn,
that the sun, when just on the horizon, subtends at our

CHAP, I.] OF THE SPHERE CF THE HEAVEXS, 39

eyes almost exactly the same, and the moon a materially
less angle, than when seen at a great altiuule in the sky,
owing to the effect of what is called parallax, to be ex-
plained presently, ^j _

(48.) After what has been said of the small extent of
the atmosphere in comparison of the mass of the earth,
we shall have little hesitation in admitting those lumina-
ries which people and adorn the sky, and which, while
they obviously form no part of the earth, and receive no
support from it, are yet not borne along at random like
clouds upon the air, nor drifted by the winds, to be ex-
ternal to our atmosphere. As such we have considered
them while speaking of their refractions — as existing in
the immensity of space beyond, and situated, perhaps,
for any thing we can perceive to the contrary, at enor-
mous distances from us and from each other.

(49.) Could a spectator exist unsustained by the earth,
or any solid support, he would see around him at one
view the whole contents of space — the visible consti-
tuents of the universe : and, in the absence of any means
of judging of their distances from him, would refer them,
in the directions in which they were seen from his sta-
tion, to the concave surface of ah imaginary sphere,
having his eye for a centre, and its surface at some vast
indeterminate distance. Perhaps he might judge those
which appear to him large and bright, to be nearer to
him than the smaller and less brilliant ; but, independent
of other means of judging he would have no warrant for
this opinion, any more than for the idea that all were
equidistant from him, and really arranged on such a
spherical surface. Nevertheless, there would be no
impropriety in his referring their places, geometrically
speaking, to those points of such a purely imaginary
sphere, which their respective visual rays intersect ; and
there would be much advantage in so doing, as by that
means their appearance and relative situation could be
accurately measured, recorded, and mapped down. The
objects in a landscape are at every variety of distance
from the eye, yet we lay them all down in a picture on
one plane, and at one distance, in their actual appareyit
proportions, and the likeness is not taxed with incorrect-

40 A TREATISE ON ASTRONOMY. [cHAP. I.

ness, though a man in the foreground should be repre-
sented hirger than a mountain in the distance. So it is
to a spectator of the heavenly bodies pictured, projected,
or mapped down on that imaginary sphere we call tlie
sky or heaven. Thus, we may easily conceive that the
moon, which appears to us as large as the sun, though
less bright, may owe that apparent equality to its greater
proximity, and may be really much less ; while both the
moon and sun may only appear larger and brighter than
the stars, on account of the remoteness of the latter.

(50.) A spectator on the earth's surface is prevented,
by tlie great mass on which he stands, from seeing into
all that portion of space which is below him, or to see
which he must look in any degree downwards. It is
true that, if his place of observation be at a great eleva-
tion, the dip of the horizon will bring within the scope
of vision a little more than a hemisphere, and refraction,
wherever he may be situated, will enable him to look,
as it were, a little round the corner; but the zone thus
9.dded to his visual range can hardly ever, unless in very
extraordinary circumstances,* exceed a couple of degrees
in breadth, and is always ill seen on account of the va-
pours near the horizon. Unless, then, by a change of
his geographical situation, he should shift his horizon
(which is always a plane touching the spherical con-
vexity of the earth at his station) ; or unless, by some
movements proper to the heavenly bodies, they should
of themselves come above his horizon ; or, lastly, un-
less, by some rotation of the earth itself on its centre,
the point of its surface which he occupies should be
carried round, and presented towards a different region
pf space ; he would never obtain a sight of almost one

*Such as the following, for instance: The late Mr. Sadler, the cele^
brated aeronaut, ascended in a balloon from Dublin at about 2 o'clock in
the afternoon, and was wafted across the channel. About sunset he ap-
proached the English coast, when the balloon descended near the surface
of the sea. By this time the sun was set, and the shades of evening began
to close in. He threw out nearly all his ballast, and suddenly sprung
upwards to a great height, and by so doing witnessed the whole pheno-
menon of a western sunrise. He subsequently descended in Wales, and
witnessed a second sunset on the same evening. I have this anecdote
from Dr. Lardner, who was present at his ascent, and read his owp acr
cpuntofthe voyvigQ.^^ Author.

CHAP. 1.] CHANGE OF LOCAL SITUATION. 41

half the objects external to our atmosphere. But if any
of these cases be supposed, more, or all, may come into
view accordincf to the circumstances.

(51.) A traveller, for example, shifting his locality on
our globe, will obtain a view of celestial objects invisible
from his original station, in a way which may be not in-
aptly illustrated by comparing him to a person standing
in a park close to a large tree. The massive obstacle
presented by its trunk cuts off his view of all those parts
of the landscape which it occupies as an object; but by
walking round it a complete successive view of the
whole panorama may be obtained. Just in the same
way, if we set off from any station, as Loudon, and
travel southwards, we shall not fail to notice that many
celestial objects which are never seen from London
come successively into view, as if rising up above the
horizon, night after night, from the south, although it is
in reality our horizon, which, travelling with us south-
wards round the sphere, sinks in succession beneath
them. The novelty and splendour of fresh constella-
tions thus gradually brought into view in the clear calm
nights of tropical climates, in long voyages to the south,
is dwelt upon by all who have enjoyed this spectacle,

^

*

and never fails to impress itself on the recollection
among the most delightful and interesting of the asso-
ciations connected with extensive travel. A glance at

d2

42 A TREATISE ON ASTRONOMY. [ciIAP. I.

the accompLinying figure, exhibiting three successive
stations of a traveller, A, B, C, with the horizon cor-
responding to each, will place this pi'ocess in clearer
evidence than any description.

(52.) Again : suppose the earth itself to have a mo-
tion of rotation on its centre. It is evident that a spec-
tator at rest (as it appears to him) on any part of it will,
imperceived by himself, be carried round with it : un-
perceived, we say, because his horizon will constantly
contain, and be limited by, the same terrestrial objects.
He will have the same landscape constantly before his
eyes, in which all the familiar objects in it, which serve
him for landmarks and directions, retain, with respect
to himself or to each other, the same invariable situa-
tions. The perfect smoothness and equality of the
motion of so vast a mass, in which every object he sees
around him participates alike, will (art. 1 5) prevent his
entertaining any suspicion of his actual change of place.
Yet, with respect to external objects, — that is to say,
all celestial ones which do not participate in the sup-
posed rotation of the earth, — his horizon will have been
all the while shifting in its relation to them, precisely as
in the case of our traveller in the foregoing article. Re-
curring to the figure of that article, it is evidently the
same thing, so far as their visibility is concerned,
whether he has been carried by the earth's rotation suc-
cessively into the situations A, B, C ; or whether, the
earth remaining at rest, he has transferred himself per-
sonally along its surface to those stations. Our spectator
in the park will obtain precisely the same view of the
landscape, whether he walk round the tree, or whether
we suppose it sawed off", and made to turn on an upright
pivot, while he stands on a projecting step attached to it,
and allows himself to be carried round by its motion.
The only difterence will be in his view of the tree it-
pelf, of which, in the former case, he will see every part,
but, in the latter, only that portion of it which remains
constantly opposite to him, and immediately under hia
eye.

(53.) By such a rotation of the earth, then, as we
have supposed, the horizon of a stationary spectator will

CHAP. 1.] DIURNAL ROTATION OF THE EARTH. 43

be constantly depressing itself below those objects which
lie in that region of space towards which the rotation is
carrying him, and elevating itself above those in the op-
posite quarter ; admitting into view the former, and suc-
cessively hiding the latter. As the horizon of every such
spectator, however, appears to him motionless, all such
changes will be referred by him to a motion in the objects
themselves so successively disclosed and concealed. In
place of his horizon approaching the stars, therefore, he
will judge the stars to approach his horizon ; and when it
passes over and hides any of them, he will consider
them as having sunk below it, or set ; while those it has
just disclosed, and from which it is receding, will seem
to be rising above it.

(54.) If we suppose this rotation of the earth to con-
tinue in one and the same direction, — that is to say, to be
performed round one and the same axis, till it has com-
pleted an entire revolution, and come back to the position
from which it set out when the spectator began his obser-
vations,— it is manifest that every thing will then be in
precisely the same relative position as at the outset : all
the heavenly bodies will appear to occupy the same
places in the concave of the sky which they did at that
instant, except such as may have' actually moved in the
interim ; and if the rotation still continue, the same phe-
nomena of their successive rising and setting, and return
to the same places, will continue to be repeated in the
same order, and (if the velocity of rotation be uniform)
in equal intervals of time, ad infinitum.

(55,) Now, in this we have a lively picture of that
grand phenomenon, the most important beyond all com-
parison which nature presents, the daily rising and setting
of the sun and stars, their progress through the vault of
the heavens, and their return to the same apparent places
at the same hours of the day and night. The accom-
plishment of this restoration in the regular interval of
twenty-four hours, is the first instance we encounter of
that great law of periodicity,* which, as we shall see,
pervades all astronomy ; by which expression we under-

* rTjeioJof, a going round, a circulation or revolution

44 A TREATISE ON ASTRONOMY. [c'HAP. I.

Stand the continual reproduction of tlie same phenomena,
in the same order, at equal intervals ol' time.

(56.) A free rotation of the earth round its centre, if it
exist and be performed in consonance with the same me-
chanical laws which obtain in the motions of masses of
matter under our immediate control, and within our ordi-
nary experience, must be such as to satisfy two essential
conditions. It must be invariable in its direction with
respect to the sphere itself, and uniform in its velocity.
The rotation must be performed round an axis or diame-
ter of the sphere, whose poles, or extremities, where it
meets the surface, correspond always to the same points
on the sphere. Modes of rotation of a solid body under
the influence of external agency are conceivable, in which
the poles of the imaginary line or axis about which it is
at any moment revolving shall hold no fixed places on the
surface, but shift upon it every moment. Such changes,
however, are inconsistent with the idea of a rotation of
a body of regular figure about its axis of symmetry, per-
formed in free space, and without resistance or obstruc-
tion from any surrounding medium. The complete ab-
sence of such obstructions draws with it, of necessity,
the strict fulfilment of the two conditions above men-
tioned.

(57.) Now, these conditions are in perfect accordance
with what we observe, and what recorded observation
teaches us in respect of the diurnal motions of the hea-
venly bodies. We have no reason to believe, from his-
tory, that any sensible change has taken place since the
earliest ages in the interval of time elapsing between two
successive returns of the same star to the same point of
the sky ; or, rather, it is demonstrable from astronomical
records that no such change has taken place. And with
respect to the other condition, — the permanence of the
axis of rotation, — 'the appearances which any alteration
in that respect must produce, would be marked, as we
shall presently show, by a corresponding change of a
very obvious kind in the apparent motions of the stars ;
which, again, history decidedly declares them not to have
undergone.

(58.) But, before we proceed to examine more in de-

CHAP. I.J APPARENT DIURNAL MOTION. 45

tail how the liypothesis of the rotation of tlie earth about
an axis accords with the plienoniena which the diurnal
motion of the heavenly bodies ofl'ers to our notice, it will
be proper to describe, with precision, in what that diur-
nal motion consists, and how far it is participated in
by them all ; or whether any of them form exceptions,
wholly or partially, to the common analogy of the rest.
We will, therefore, suppose the reader to station himself,
on a clear evening, just after sunset, when the first stars
begin to appear, in some open situation whence a good
general view of the heavens can be obtained. He will
then perceive, above and around him, as it were, a vast
concave hemispherical vault, beset with stars of various
magnitudes, of which the brightest only will first catch
his attention in the twilight ; and more and more will
appear as the darkness increases, till the Avhole sky is
overspangled with them. When he has awhile admired
the calm magnificence of this glorious spectacle, the
theme of so much song, and of so much thought, — a
spectacle which no one can view without emotion, and
without a longing desire to know something of its na-
ture and purport, — let him fix his attention more particu-
larly on a few of the most brilliantstars, such as he can-
not fail to recognise affuin without mistake after looking
away from them for some time, and let him refer their ap-
parent situations to some surrounding objects, as build-
ings, trees, &c., selecting purposely such as are in dif-
ferent quarters of his horizon. On comparing them a^ain
with their respective points of reference, after a moderate
interval, as the night advances, he will not fail to per-
ceive that they have changed their places, and advanced,
as by a general movement, in a westward direction ;
those towards the eastern quarter appearing to rise or re-
cede from the horizon, while those which lie towards the
west will be seen to approach it ; and, if watched long
enough, will, for the most part, finally sink beneath it,
and disappear ; while others, in the eastern quarter, will
be seen to rise as if out of the earth, and, joining in the
general procession, will take their course with the rest
towards the opposite quarter.

(59.) If he persists for a considerable time in watch^-

46 A TREATISE ON ASTRONOMY. [cHAP. I.

ing their motions, on tlie same or on several successive
nights, he will perceive that each star appears to describe,
as far as its course lies above the horizon, a circle in the
sky ; that the circles so described are not of the same
magnitude for all the stars ; and that those described by
different stars differ greatly in respect of the parts of
them Avhich lie above the horizon, some, which lie to-
wards the quarter of the horizon which is denominated
the South,* only remain for a short time above it, and dis-
appear, after describing in sight only the small upper seg-
ment of their diurnal circle ; others, which rise between
the sovith and east, describe larger segments of their cir-
cles above the horizon, remain proportionally longer in
sight, and set precisely as far to the westward of south
as they rose to the eastward ; while such as rise exactly
in the east remain just twelve hours visible, describe a
semicircle, and set exactly in the west. With those,
again, which rise between the east and north, the same
law obtains ; at least, as far as regards the time of their
remaining above the horizon, and the proportion of the
visible segment of their diurnal circles to their whole cir-
cumferences. Both go on increasing ; they remain in
view more than twelve hours, and their visible diurnal
arcs are more than semicircles. But the magnitudes of
the circles themselves diminish, as we go from the east,
northward ; the greatest of all the circles being described
by those which rise exactly in the cast point. Carrying
his eye farther northwards, he will notice, at length, stars
which, in their diurnal motion, just graze the horizon at
its north point, or only dip below it for a moment ; while
others never reach it all, but continue always above it,
revolving in entire circles round one point, called the
POLE, which appears to be the common centre of all
their motions, and which alone, in the whole heavens,
may be considered immovable. Not that this point is
marked by any star. It is a purely imaginary centre ;
but there is near it one considerably bright star, called
the Pole Star, which is easily recognised by the very

* We suppose our observer to be stationed in some northern latitude ;
BOme where in Europe, for example.

CHAP. I.] APPARENT DIURNAL MOTION. 47

small circle it describes : so small, indeed, that, without
paying particular attention, and referring its position very
nicely to some fixed mark, it may easily be supposed at
rest, and be, itself, mistaken for the common centre about
which all the others in that region describe their circles ;
or it may be known by its configuration with a very
splendid and remarkable constellation or group of stars,
called by astronomers the Great Bear.

(60.) He will further observe that the apparent rela-
tive situations of all the stars among one another is not
changed by their diurnal motion. In Avliatever parts of
their circles they are observed, or at whatever hour of the
night, they form with each other the same identical groups
or configurations, to which the name of constellations
has been given. It is true, that, in different parts of their
course, these groups stand differently with respect to the
horizon ; and those towards the north, when in the course
of their diurnal movement they pass alternately above and
below that common centre of motion described in the last
article, become actually inverted with respect to the hori-
zon, while, on the other hand, they always turn the same
points towards the pole. In short, he will perceive that
the whole assemblage of stars visible at once, or in suc-
cession, in the heavens, may be regarded as one great
constellation, which seems to revolve with a uniform mo-
tion, as if it formed one coherent mass ; or as if it were at-
tached to the internal surface of a vast hollow sphere,
having the earth, or rather the spectator in the centre, and
turning round an axis inclined to his horizon, so as to pass
through that fixed point or pole already mentioned.

(61.) Lastly, he will notice, if he have patience to
outwatch a long winter's night, commencing at the earli-
est moment when the stars appear, and continuing till
morning twilight, that those stai-s which he observed set-
ting in the west have again risen in the east, while those
which were rising when he first began to notice them
have completed their course, and are now set ; and that
thus the hemisphere, or a great part of it, which Avas then
above, is now beneath him, and its place supplied by that
which was at first under his feet, Avhich he will thus disco-
ver to be no less copiously furnished with stars than the

48 A TREATISE OfJ ASTRONOMY. [cHAP. t.

Other, and bespangled with groups no less permanent and
distinctly recognisable. Thus he will learn that the great
constellation we have above spoken of as revolving round
the pole is co-extensive with the whole surface of the
sphere, being in reality nothing less than a universe of
luminaries surrounding the earth on all sides, and brought
in succession before his view, and referred (each lumina-
ry according to its own visual ray or direction from his
eye) to the imaginary spherical surface, of which he him*
self occupies the centre. (See art. 49.)

(62.) There is, however, one portion or segment of
this sphere of which he will not thus obtain a view. As
there is a segment towards the north, adjacent to the pole
above his horizon, in which the stars never set, so there
is a corresponding segment, about which the smaller cir-
cles of the more southern stars are described, in which
they never rise. The stars which border upon the extreme
circumference of this segment just graze the southern point
of his horizon, and show themselves for a few moments
above it, precisely as those near the circumference of the
northern segment graze his northern horizon, and dip for a
moment below it, to reappear immediately. Every point
in a spherical surface has, of course, another diametrically
opposite to it ; and as the spectator's horizon divides his
sphere into two hemispheres — a superior and inferior —
there must of necessity exist a depressed pole to the south,
corresponding to the elevated one to the north, and a por-
tion surrounding it, perpetually beneath, as there is an-
other surrounding the north pole, perpetually above it.

'' Hie vertex nobis semper sublimis ; at ilium
Sub pedibus noxatra videt, manesque profundi." — Virgil.

One pole rides high, one, plimged beneath -the main,
Seeks the deep night, and Pluto's dusky reign.

(63.) To get sight of this segment, he must travel south-
wards. In so doing, a new set of phenomena come for-
ward. In proportion as he advances to the south, some
of those constellations which, at his original station, barely
grazed the northern horizon, will be observed to sink be-
low it and set ; at first remaining hid only for a very short
time,but gradually for a longerpart of the twenty-four hours.
They will continue, however, to circulate about the same

CHAP. 1.] EFFECT OF CHANGE OF LATITUDE. 'l'«^

point — that is, holding the same invariable position icith
respect to them in the concave of the heavens among the
stars ; but this point itself will become gradually depress-
ed with respect to the spectator's horizon. The axis, in
short, about which the diurnal motion is performed, will
appear to have become continually less and less inclined
to the horizon ; and by the same degrees as the northern
pole is depressed the southern will rise, and constellations
surrounding it will come into view ; at first momentarily,
but by degrees for longer and longer times in each diur-
nal revoluUon — reahzing, in short, what we have already
stated in art. 51.

(64.) If he travel continually southwards, he will at
length reach a line on the earth's surface, called the equa-
tor, at any point of which, indifferently, if he take up his
station and recommence his observations, he will find that
he has both the centres of diurnal motion in his horizon,
occupying opposite points, the northern pole having been
depressed, and the southern raised ; so that, in this geo-
graphical position, the diurnal rotation of the heavens
will appear to him to be performed about a horizontal
axis, every star describing half its diurnal circle above and
half beneath his horizon, remaining alternately visible for
twelve hours, and concealed during the same interval.
In this situation, no part of the heavens is concealed from
his successive view. In a night of twelve hours (suppo-
sing such a continuance of darkness possible at the equa-
tor) the whole sphere will have passed in review over
him — the whole hemisphere with which he began his
night's observation will have been carried down beneath
him, and the entire opposite one brought up from below.
(65.) If he pass the equator, and travel still farther
southwards, the southern pole of the heavens will become
elevated above his horizon, and the northern will sink
below it ; and the more, the farther he advances south-
wards ; and when arrived at a station as far to the south
of the equator as that from which he started was to the
north, he will find the whole phenomena of the heavens
reversed. The stars which at his original station de-
scribed their whole diurnal circles above his horizon, and
never set, now describe them entirely below it, and never

E

50 A TREATISE ON ASTROKOMY. [cHAP. I.

rise, but remain constantly invisible to him ; and vice
versa, those stars which at his former station he never
saw, he will now never cease to see.

(66.) Finally, if instead of advancing southwards from
his first station, he travel northwards, he will observe the
northern pole of the heavens to become more elevated
above his horizon, and the southern more depressed be-
low it. In consequence, his hemisphere will present a
less variety of stars, because a greater proportion of the
whole surface of the heavens remains constantly visible
or constantly invisible : the circle described by each star,
too, becomes more nearly parallel to the horizon ; and,
in short, every appearance leads to suppose that could he
travel far enough to the north, he would at length attain
a point vertically under the northern pole of the heavens,
at which none of the stars would either rise or set, but
each Avould circulate round the horizon in circles parallel
to it. Many endeavours have been made to reach this
point, which is called the north pole of the earth, but
hitherto without success ; a barrier of almost insurmount-
able difficulty being presented by the increasing rigour
of the climate : but a very near approach to it has been
made ; and the phenomena of those regions, though not
precisely such as we have described as what must subsist at
the pole itself, have proved to be in exact correspondence
with its near proximity. A similar remark applies to the
south pole of the earth, which, however, is more unap-
proachable, or, at least, has been less nearly approached,
than the north.

(67.) The above is an account of the phenomena of
the diurnal motion of the stars, as modified by different
geographical situations, not grounded on any specula-
tion, but actually observed and recorded by travellers
and voyagers. It is, however, in complete accordance
with the hypothesis of a rotation of the earth round a
fixed axis. In order to show this, however, it will be
necessary to premise a few observations on the appear-
ances presented by an assemblage of remote objects,
when viewed from different parts of a small and circum-
scribed station.

''68.) Imagine a landscape, in which a great multitude

DISTANCE OF THE STARS.

51

CHAP. I.J

of objects are placed at every variety of distance from the
beholder. If he shift his point of view, though but for
a few paces, he will perceive a very great change in the
apparent positions of the nearer objects, both with re-
spect to himself and to each other. If he advance north-
wards, for instance, near objects on his right and left,
which were, therefore, to the east and west of his
original station, will be left behind him, and appear to
have receded southwards ; some, which covered each
other at first, will appear to separate, and others to ap-
proach, and perhaps conceal each other. Remote objects,
on the contrary, will exhibit no such great and remarka-
ble changes of relative position. An object to the east
of his original station, at a mile or two distance, will

still be referred by him to the east point of his horizon,
with hardly any perceptible deviation. The reason of
this is, that the position of every object is refeiTed by lis
to the surface of an imaginary sphere of an indefinite ra-
dius, having our eye for its centre ; and, as we advance
in any direction, AB, carrying this imaginary sphere
along with us, the visual rays AP, AQ, by which ob-
jects are referred to its surface (at C, for instance), shift
their positions with respect to the line in which we
move, AB, which serves as an axis or line of reference,
and assume new positions, BPp, BQ^', revolving round
their respective objects as centres. Their intersections,
therefore, p, q, with our visual sphere, will appear to
recede on its surface, but with different degrees of an-
gular velocity in proportion to their proximity ; the
same distance of advance AB subtending a greater an-
gle, APB=cPp, at the near object P than at the remote
one Q,

53 A TREATISE ON ASTRONOMY. [cHAP. I.

(69.) This apparent angular motion of an object on
our sphere of vision,* arising from a cliange of our point
of view, is called parallax, and it is always expressed
by the angle BAP subtended at the object P by a line
joining the two points of view AB under consideration.
For it is evident that the diflerence of angular position
of P, with respect to the invariable direction ABD,
when viewed from A and from B, is the difference of
the two angles DBP and DAP ; now, DBP being the
exterior angle of the triangle, ABP is equal to the sum
of the interior and opposite, DBP=DAP + APB, whence
DBP— DAP=APB.

(70.) It follows from this, that the amount of paral-
lactic motion arising from any given change of our point
of view is, cseteris jxiribus, less, as the distance of an
object viewed is greater ; and when that distance is ex-
tremely great in comparison with the change in our point
of view, the parallax becomes insensible ; or, in other
words, objects do not appear to vary in situation at all.
It is on this principle, that in alpine regions visited for
the first time we are surprised and confounded at the
little progress we appear to make by a considerable
change of place. An hour's Avalk, for instance, produces
but a small parallactic change in the relative situations
of the vast and distant masses which surround us.
Whether we walk round a circle of a hundred yards in
diameter, or merely turn ourselves round in its centre,
the distant panorama presents almost exactly the same as-
pect,— we hardly seem to have changed our point of vicAV.

* The ideal sphere without us, to which we refer the places of objects,
and which we carry along with us wherever we go, is no doubt inti-
mately connected by association, if not entirely dependent on that ob-
scure perception of sensation in the retinae of our eyes, of which, even
when closed and unexcited, we cannot entirely divest them. We have
a real spherical surface within our eyes, the seat of sensation and vision,
corresponding, point for point, to the external sphere. On this the stars,
&c. are really mapped down, as we have supposed them in the text to
be, on the imaginary concave of the heavens. Wlien the whole surface
of the retinae is excited by light, habit leads us to assoc^iate it with the
idea of a real surface existing without us. Thus we become impressed
with the notion of a sky and a heaven, but the concave surface of the
retinas itself is the true seat of all visible angular dimension and angular
motion. The substitution of the retina for the Aeawens would be awkward
and inconvenient in language, but it may always be mentally made,
(See Schiller's jiretty enigma on the eye in his Turoiidot.)

CHAP. 1.] DISTANCE OF THE STARS. 53

(71.) Whatever notion, in other respects, we may
form of the stars, it is quite clear they must be im-
mensely distant. Were it not so, the apparent angular
interval between any two of them seen over head would
be much greater than wlien seen near the horizon, and
the constellations, instead of preserving the same ap-
pearances and dimensions during their whole diurnal
course, would appear to enlarge as they rise higher in
the sky, as we see a small cloud in the horizon swell
into a great overshadowing canopy when drifted by the
wind across our zenith, or as may be seen in the annex-
ed figure, where ab, AB, a b, are three difierent positions of

the same stars, as they would, if near the earth, be seen
from a spectator S, under the visual angles aSb, ASB.
No such change of apparent dimension, however, is ob-
served. The nicest measurements of the apparent an-
gular distance of any two stars inter se, taken in any
parts of their diurnal course, (after allowing for the un-
equal effects of refraction, or when taken at such times
that this cause of distortion shall act equally on both,)
manifest i2ot the slightest perceptible variation. Not
only this, but at whatever point of the earth's surface the
measurement is performed, the results are absolutely
identical. No instruments ever yet invented by man
are delicate enough to indicate, by an increase or dimi-
nution of the angle subtended, that one point of the
earth is nearer to or further from the stars than another.
(72.) The necessary conclusion from this is, that the
dimensions of the earth, large as it is, are comparatively
nothing, absolutely imperceptible, when compared with

e3

54 A TREATISE ON ASTRONOMY. [cHAP. I.

the interval which separates the stars from the earth. If
an observer walk round a circle not more than a few
yards in diameter, and from diflerent points in its cir-
cumference measure with a sextant, or other more exact
instrument adapted for the purpose, the angles PAQ,
PBQ, PCQ, subtended at those stations by two well
defined points in his visible horizon, PQ, he will at once

be advertised, by the difference of the results, of his
change of distance from them arising from liis change
of place, although that difference may be so small as to
produce no change in tlieir general aspect to his unas-
sisted sight. This is one of the innumerable instances
where accurate measurement obtained by instrumental
means places us in a totally different situation in respect
to matters of fact, and conclusions thence deducible,
from what we should hold, were we to rely in all cases
on the mere judgment of the eye. To so great a nicety
have such observations been carried by the aid of an
instrument called a theodolite, that a circle of the dia-
meter above mentioned may thus be rendered sensible,
may thus be detected to have a size, and an ascertainable
place, by reference to objects distant by fully 100,000
times its own dimensions. Observations, differing, it is
true, somewhat in method, but identical in principle,
and executed with nearly as much exactness, have been
applied to the stars, and with a result such as has been
already stated. Hence it follows, incontrovertibly, that
the distance of the stars from the earth cannot be so

CHAP. 1.3 DISTANCE OF THE STARS. 55

small as 100,000 of the earth's diameters. It is, indeed,
incomparably greater ; for we shall hereafter find it fully
demonstrated that the distance just named, immense as it
may appear, is yet much underrated.

(73.) From such a distance, to a spectator with our
faculties, and furnished with our instruments, the earth
would be imperceptible ; and, reciprocally, an object of
the earth's size, placed at the distance of the stars, would *
be equally undiscernible. If, therefore, at the point on
which a spectator stands, we draw a plane touching the
globe, and prolong it in imagination till it attain the
region of the stars, and through the centre of the earth
conceive another plane parallel to the former, and co- -
extensive with it, to pass ; these, although separated
throughout their whole extent by the same interval, viz.
a semi-diameter of the earth, will yet, on account of the
vast distance at which that interval is seen, be confound-
ed together, and undistinguishable from each other in the
region of the stars, when viewed by a spectator on the
earth. The zone they there include will be of evanescent
breadth to his eye, and will only mark out a great circle in
the heavens, Avhich, like the vanishing point in perspec-
tive to which all parallel lines in a picture appear to
converge, is, in fact, the vanishing line to which all
planes parallel to the horizon offer a similar appearance
of ultimate convergence in the great panojxmia of nature.

(74.) The two planes just described are termed, in
astronomy, the sensible and rational horizon of the ob-
server's station ; and the great circle in the heavens which
marks their vanishing line, is also spoken of as a circle
of the sphere, under the name of the celestial horizon,
or simply the horizon.

From what has been said (art. 72) of the distance
of the stars, it follows, that if we suppose a spectator
at the centre of the earth to have his view bounded by
the rational horizon, in the same manner as that of a
corresponding spectator on the surface is by his sensible
horizon, the two observers will see the same stars in the
same relative situations, each beholding that entire he-
misphere of the heavens which is above the celestial
horizon, coiTesponding to their common zenith.

58 A TREATISE ON AfiTUONOIirY, [cHAP. I.

(75.) Now, so far as appearances g-o, it is clearly the
same thing whether the heavens, that is, all space, with
its contents, revolve round a spectator at rest in the earth's
centre, or whether that spectator simply turn round in the
opposite direction in his place, and view them in suc-
cession. The aspect of the heavens, at every instant, as
referred to his horizon (which must be supposed to turn
with him), will be the same in both supposition-3. And
since, as has been shown, appearances are also, so far as
the stars are concerned, the same to a spectator on the sur-
face as to one at the centre, it follows that, whether we sup-
pose the heavens to revolve without the earth, or the earth
within the heavens, in the opposite direction, the diurnal
phenomena, to all its inhabitants, will be no way different.
(76.) The Copernican astronomy adopts the latter as
the true explanation of these phenomena, avoiding there-
by the necessity of otherwise resorting to the cumbrous
mechanism of a solid but invisible sphere, to which the
stars must be supposed attached, in order that they may
be carried round the earth without derangement of their
relative situations inter se. Such a contrivance would,
indeed, suffice to explain the diurnal revolution of the
stars, so as to " save appearances ;" but the movements of
the sun and moon, as well as those of the planets, are in-
compatible with such a supposition, as Avill appear when
we come to treat of these bodies. On the other hand, that
a spherical mass of moderate dimensions (or, rather,
when compared with the surrounding and visible universe,
of evanescent magnitude), held by no tie, and free to move
and to revolve, should do so, in conformity with those
general laws which, so far as we know, regulate the mo-
tions of all material bodies, is so far from being a postu-
late difficult to be conceded, that the wonder would rather
be should the fact prove otherwise. As a postulate, there-
fore, we shall henceforth regard it; and as, in the pro-
gress of our work, analogies offer themselves in its sup-
port from what we observe of other celestial bodies, we
shall not fail to point them out to the reader's notice.
Meanwhile, it will be proper to define a variety of terms
which will be continually employed hereafter.

(77.) Definition 1. The axis of the earth is that di'

CHAP. I.] DEFINITIONS. 57

ameter about which it revolves, with a uniform motion,
from west to east ; performing one revolution in the in-
terval which elapses between any star leaving a certain
point in the heavens, and returning to the same point

again.

(78.) Def. 2. The jioles of the earth are the points
where its axis meets its surface. The North Pole is that
nearest to Europe ; the South Pole that most remote from it.

(79.) Def. 3. The sphere of the heavens, or the sphere
of the stars, is an imaginary spherical surface of infinite
radius, and having the centre of the earth, or, which
comes to the very same thing, the eye of any spectator
on its surface, for its centre. Every point in this sphere
may be regarded as the vanishing point of a system of
lines parallel to that radius of the sphere which passes
through it, seen in perspective from the earth ; and any
great circle on it, as the vanishing line of a system of
planes parallel to its own. This mode of conceiving such
points and circles has great advantages in a variety of cases.

(80.) Def. 4. The zenith and 7iadir* are the two points
of the sphere of the heavens, vertically over tlie specta-
tor's head, and vertically under his feet ; they are, there-
fore, the vanishing points of all lines mathematically pa-
rallel to the direction of a plumb-line at his station. The
plumb-line itself is, at every point of the earth, perpen-
dicular to its spherical surface : at no two stations, there-
fore, can the actual directions of two plumb-lines be re-
garded as mathematically parallel. They converge to-
wards the centre of the earth : but for very small intervals
(as in the area of a building-r-in one and the same town,
&;c.) the difference from exact parallelism is so small, that
it may be practically disregarded. An interval of a mile
corresponds to a convergence of plumb-lines amounting
to about 1 minute. The zenith and nadir are the poles
of the celestial horizon ; that is to say, points 90° distant
from every point in it. The celestial horizon itself is
the vanishing line of a system of planes parallel to the
sensible and rational horizon.

* From Arabic worjs. Nadir corresponds evidently to the German
nieder (down)

58 A TREATISE ON ASTRONOMY. [cHAP. I.

(81.) Def. 5. Vertical circles of the sphere are ^eat
circles passing through the zenith and nadir, or great cir-
cles perpendicular to the horizon. On these are mea-
sured the altitudes of objects above the horizon — the
complements to which are their zenith distances.

(82.) Def. 6. The poles of the heavens are the points
of the sphere to which the earth's axis is directed ; or
the vanishing points of all lines parallel thereto.

(83.) Def. 7. The earth'' s equator is a great circle on
its surface, equidistant from its poles, dividing it into
two hemispheres — a northern and a southern ; in the
midst of which are situated the respective poles of the.
earth of those names. The plane of the equator is,
therefore, a. plane perpendicular to the earth's axis, and
passing through its centre. The celestial equator is a
great circle of the heavens, marked out by the indefinite
extension of the plane of the terrestrial, and is the vanish-
ing line of all planes parallel to it. This circle is called
by astronomers the equinocticd.

(84.) Def. 8. The terrestrial meridian of a station
on the earth's surface is a great circle passing through
both the poles and through the place. When its plane
is prolonged to the sphere of the heavens, it marks out
the ce/es^za/wienV/iVm of a spectator stationed at that place.
When we speak of the meridian of a spectator, we intend
the celestial meridian, which is a vertical circle passing
through the poles of the heavens.

The plane of the meridian is the plane of this circle,
and its intersection with the sensible horizon of the spec-
tator is called a meridian line, and marks the north and
south points of his horizon.

(85.) Def. 9. Jlzimuth is the angular distance of a
celestial object from the north or south point of the hori-
zon (according as it is the north or south pole which is
elevated), when the object is referred to the horizon by
a vertical circle ; or it is the angle comprised between
two vertical planes — one passing through the elevated
pole, the other through the object. The altitude and
azimuth of an object being known, therefore its place in
the visible heavens is determined. For their simultane^
ous measurement, a peculiar instrument has been ima»

Chap, i.] latitude and loxgitude. 69

gined, called an altitude and azimuth instrument, which
will be described in the next chapter.

(86.) Def. 10. The latitude of a place on the earth's
surface is its angular distance from the equator, measured
on its own terrestrial meridian : it is reckoned in degrees,
minutes, and seconds, from 0 up to 90°, and northwards
or southwards according to the hemisphere the place lies
in. Thus, the observatory at Greenwich is situated in
51° 28' 40" north latitude. This definition of latitude, it
will be observed, is to be considered as only temporary.
A more exact knowledge of the physical structure and
figure of the earth, and a better acquaintance with the
niceties of astronomy, will render some modification of its
terms, or a different manner of considering it, necessary.
(87.) Def. 11. Parallels of latitude are small circles
on the earth's surface parallel to the equator. Every
point in such a circle has the same latitude. Thus, Green-
wich is said to be situated in the parallel of 51° 28' 40".
(88.) Def. 12. The /o«g"i7w(Ze of a place on the earth's
surface is the inclination of its meridian -to that of some
fixed station referred to as a point to reckon from. Eng-
lish astronomers and geographers use the observatory at
Greenwich for this station ; foreigners, the principal ob-
servatories of their respective nations. Some geographers
have adopted the island of Ferro. Hereafter, when we
speak of longitude, we reckon from Greenwich. The
longitude of a place is, therefore, measured by the arc of
the equator intercepted between the meridian of a place
and that of Greenwich ; or, which is the same thing, by
the spherical angle at the pole included between these
meridians.

As latitude is reckoned north or south, so longitude
is usually said to be reckoned west or east. It would
add greatly, however, to systematic regularity, and tend
much to avoid confusion and ambiguity in computations,
were this mode of expression abandoned, and longitudes
reckoned invariably ivestu'ard from their origin round
the whole circle from 0 to 360°. Thus the longitude
of Paris is, in common parlance, either 2° 20' 22" east,
or 357° 39' 38" west of Greenwich. But, in the sense
on which we shall henceforth use and recommend others

60 A TREATISE ON ASTRONOMY. [CHAP. I.

to use the term, the latter is its proper designation.
Longitude is also reckoned in time at the rate of 24 h.
for 360°, or 15° per hour. In this system the longitude
of Paris is 23h. 50m. 38|s.

(89.) Knowing the longitude and latitude of a place,
it may be laid down on an artificial globe ; and thus a
map of the earth may be constructed. Maps of particu-
lar countries are detached portions of this general map,
extended into planes ; or, rather, they are representations
on planes of such portions, executed according to certain
conventional systems of rules, called projections, the
object of which is either to distort as little as possible
the outlines of countries from what they are on the globe
— or to establish easy means of ascertaining, by inspec-
tion or graphical measurement, the latitudes and longi-
tudes of places which occur in them, without referring
to the globe or to books — or for other peculiar uses. See
chap. III.

(90.) A globe, or general map of the heavens, as well
as charts of particular parts, may also be constructed,
and the stars laid down in their proper situations rela-
tive to each other, and to the poles of the heavens and
the celestial equator. Such a representation, once made,
will exhibit a true appearance of the stars as they pre-
sent themselves in succession to every spectator on the
surface, or as they may be conceived to be seen at once
by one at the centre of the globe. It is, therefore, in-
dependent of all geographical localities. There will
occur in such a representation neither zenith, nadir, nor
horizon — neither east nor west points ; and although
great circles may be drawn on it from pole to pole, cor-
responding to terrestrial meridians, they can no longer,
in this point of view, be regarded as the celestial meri-
dians of fixed points on the earth's surface, since, in
the course of one diurnal revolution, every point in it i
passes beneath each of them. It is on account of this
change of conception, and with a view to establish a
complete distinction between the two branches of Geo-
graphy and Uranography,* that astronomers have
odopted different terms (viz. declination, and right
* Tt), the earth; yea^Mv, to describe or represent: ovfuve;, the heavens.

CHAP. I.J LATITUDE AND LONGITUDE. 61

ascension) to represent those arcs in the heavens which
correspond to latitudes and longitudes on the earth. It
is for this reason that they term the equator of the hea-
vens the equinoctial ; that what are meridians on the
earth are called hour circles in the heavens, and the
angles they include between them at the poles are called
hour angles. All this is convenient and intelligible ;
and had they been content with this nomenclature, no
confusion could ever have arisen. Unluckily, the early
astronomers have employed also the words latitude and
longitude in their uranography, in speaking of arcs of
circles not corresponding to those meant by the same
words on the earth, but having reference to the motion
of the sun and planets among the stars. It is now too
late to remedy this confusion, which is ingrafted into
every existing work on astronomy : we can only regret,
and warn the reader of it, that he may be on his guard
when, at a more advanced period of our work, we
shall have occasion to define and use the terms in their
celestial sense, at the same time urgently recommending
to future writers the adoption of others in their places.

(91.) As terrestrial longitudes reckon from an assumed
fixed meridian, or from a determinate point on the equa-
tor ; so right ascensions in the heavens require some
determinate hour circle, or some known point in the
equinoctial, as the commencement of their reckoning, or
their zero point. The hour circle passing through some
remarkably bright star might have been chosen ; but there
would have been no particular advantage in this ; and
astronomers have adopted, in preference, a point in the
equinoctial called the equinox, through which they sup-
pose the hour circle to pass, from which all others are
reckoned, and which point is itself the zero point of all
right ascensions, counted on the equinoctial.

The right ascensions of celestial objects are always
reckoned eastward from the equinox, and are estimated
either in degrees, minutes, and seconds, as in the case
of terrestrial longitudes, from 0° to 360°, which com-
pletes the circle ; or, in time, in hours, minutes, and
seconds, from 0 h. to 24 h. The apparent diurnal motion
of the heavens being contrary to the real motion of the

F

62 A TREATISE ON ASTRONOMY. [cHAP. h

earth, this is in conformity with the westward reckon-
ing of longitudes. (Art. 87.)

(92.) Sidereal time is reckoned by the diurnal motion
of the stars, or rather of that point in the equinoctial
from which right ascensions are reckoned. This point
may be considered as a star, though no star is, in fact,
there ; and, moreover, the point itself is liable to a cer-
tain slow variation, — ^so slow, however, as not to affect,
perceptibly, the interval of any two of its successive
returns to the meridian. This interval is called a side-
real day, and is divided into 24 sidereal hours, and these
again into minutes and seconds. A clock whicli marks
sidereal time, i. e. which goes uniformly at such a rate
as always to show 0 h. Om. Os. when the equinox comes
on the meridian, is called a sidereal clock, and is an in-
dispensable piece of furniture in every observatory.

(93.) It remains to illustrate these descriptions by
reference to a figure. Let C be the centre of the earth,

NCS its axis ; then are N and S its poles; EQ its eqiia-'
tor; AB the parallel of latitude of the station A on its
surface ; AP parallel to SON, the direction in which an
observer at A will see the elevated pole of the heavens ;
and AZ, the prolongation of the terrestrial radius CA,

CHAP. I.] DEFINITIONS EXEMPLIFIED. 63

that of his zenith. NAES will be his meridian ; NGS
that of some tixed station, as Greenwich; and GE, or
the spherical angle GNE, his longitude, and EA his la-
titude. Moreover, if ns be a plane touching the surface
in A, this will be his sensible horizon ; wAs marlied on
that plane by its intersection with his meridian will be
his meridian line, and n and s the north and south points
of his horizon.

(94.) Again, neglecting the size of the earth, or con-
ceiving him stationed at its centre, and referring every
thing to his rational horizon ; let the annexed figure
represent the sphere of the heavens ; C the spectator ;
Z his zenith ; and N his nadir ; then will HAO a great
circle of the sphere, whose poles are ZN, be his celes-
tial horizon ; Pp the elevated and depressed poles of

the heavens ; HP the altitude of the pole, and HPZEO
his meridian; ETQ, a great circle perpendicular to Pp,
will be the equinoctial ; and ifT represent the equinox,
'V T will be the 7'ight ascension, TS the declination, and
PS the polar distance of any star or object S, referred to
the equinoctial by the hour circle PST/j; and BSD will
be the diurnal circle it will appear to describe about the
pole. Again, if we reier it to the horizon by the vertical
circle ZSA, HA will be its azimuth, AS its altitude, and
ZS its zenith distance. H and O are the north and
pouth, and ew the east and west points of his horizon,

64 A TKEATISK ON ASTRONOMY. [cHAP. I.

or of the heavens. Moreover, if HA, Oo, be small cir-
cles, or parallels of declination, touching the horizon in
its north and south points, HA will be the circle of per-
petual apjiurition, between which and the elevated pole
the stars never set; Oo that of perpetual occultation,
between which and the depressed pole they never rise.
In all the zone of the heavens between HA and Oo,
they rise and set, any one of them, as S, remaining above
the horizon, in that part of its diurnal circle represented
by ABA, and below it throughout all the part represented
by AD a. It will exercise the reader to construct this
figure for several different elevations of the pole, and foi
a variety of positions of the star S in each. The fol-
lowing consequences result from these definitions, and
are propositions which the reader will readily bear in
mind : —

(95.) The altitude of the elevated pole is equal to the
latitude of the spectator's geographical station. For,
comparing the figures of arts. 93 and 94, it appears that
the angle PAZ, between the pole and zenith, in the one
figure, which is the co-altitude (complement to 90° of the
altitude) of the pole, is equal to the angle NCA in the
other ; CN and AP being parallels whose vanishing point
is the pole. Now, NCA is the co-latitude of the plane A.

(96.) The same stars, in their diurnal revolution, come
to the meridian, successively, of every place on the globe
once in twenty-four sidereal hours. And, since the di-
urnal rotation is uniform, the interval, in sidereal time,
which elapses between the same star coming upon the
meridians of two difl^'erent places is measured by the dif-
ference of longitudes of the places.

(97.) Vice versa — the interval elapsing between two
different stars coming on the meridian of one and the
Same place, expressed in sidereal time, is the measure of
the difference of right ascensions of the stars.

This explains the reason of the double division of the
equator and equinoctial into degrees and hours.

(98.) The equinoctial intersects the horizon in the east
and west points, and the meridian in a point whose alti-
tude is equal to the co-latitude of the place. Thus, at

CHAP. I.] STARS VISIBLE BY DAY. 65

Greenwich, the altitude of the intersection of the equi-
noctial and meridian is 38° 31' 20".

(99.) All the heavenly bodies culminate (i. e. come to
their greatest altitudes) on the meridian ; which is, there-
fore, the best situation to observe them, being least con-
fused by the inequalities and vapours of the atmosphere,
as well as least displaced by refraction.

(100.) All celestial objects within the circle of perpe-.
tual apparition come twice on the meridian, above the hori-
zon, in every diurnal revolution ; once above and once
beloiv the pole. These are called their iqyper and lower
culminations.

(101.) We shall conclude this chapter by calling the
reader's attention to a fact, which, if he now learn for the
first time, will not fail to surprise him, viz. that the stars
continue visible through telescopes during the day as well
as the night ; and that, in proportion to the power of the
instrument, not only the largest and brightest of them,
but even those of inferior lustre, such as scarcely strike
the eye at night as at all conspicuous, are readily found
and followed even at noonday, — unless in that part of the
sky which is very near the sun, — by those who possess the
means of pointing a telescope accurately to the proper
places. Indeed, from the bottoms of deep narrow pits, such
as a well, or the shaft of a mine, such bright stars as pass
the zenith may even be discerned by the naked eye ; and
we have ourselves heard it stated by a celebrated optician,
that the earliest circumstance which drew his attention
to astronomy was the regular appearance, at a certain
hour, for several successive days, of a considerable star,
through the shaft of a chimney.

f3

66 A TREATISE ON ASTKONOMY. [cHAP. II

CHAPTER II.

Of the Nature of astronomical Instruments and Observations in general —
Of sidereal and solar Time — Of the Measurement of Time — Clocks,
Chronometers, the Transit [nstrnment — Of the Measurement of angular
Intervals — Application of the Telescope to Instruments destined to that
Purpose — Of the Mural Circle — Fixation of polar and horizontal points
— The Level — Plumb-line — Artificial Horizon — Collimator — Of com-
pound Instruments with co-ordinate Circles, the Equatorial — Altitude
and Azimuth Instrument — Of the Sextant and reflecting Circle — Princi-
ple of Repetition.

(102.) Our first chapter has been devoted to the
acquisition chiefly of preliminary notions respecting the
globe Ave inhabit, its relation to the celestial objects which
surround it, and the physical circumstances under which
all astronomical observations must be made, as well as to
provide ourselves with a stock of technical words of 7nost
frequent and familiar use in the sequel. We might now
proceed to a more exact and detailed statement of the
facts and theories of astronomy ; but in order to do this
with full effect, it will be desirable that the reader be
made acquainted with the principal means which astrono-
mers possess, of determining, with the degree of nicety
their theories require, the data on which tliey ground their
conclusions ; in other words, of ascertaining by measure-
ment the apparent and real magnitudes with which they
are conversant. It is only when in possession of this
knowledge that he can fully appreciate either the truth of
the theories themselves, or the degree of reliance to be
placed on any of their conclusions antecedent to trial ;
since it is only by knowing what amount of error can
certainly be perceived and distinctly measured, that he
can satisfy himself whether any theory ofiers so close an
approximation, in its numerical results, to actual phe-
nomena, as will justify him in receiving it as a true repre-
sentation of nature.

(103.) Astronomical instrument-making may be justly
regarded as the most refined of the mechanical arts, and
that in which the nearest approach to geometrical preci-
sion is required, and has been attained. It may be thought

CHAP. II.] PRACTICAf. DIFFICULTIES. 67

an easy thing, by one un^icquaintcd with the niceties re-
quired, to turn a circle in metal, to divide its circumfe-
rence into 360 equal parts, and these again into smaller sub-
divisions,— to place it accurately on its centre, and to ad-
just it in a given position ; but practically it is found to be
one of the most difficult. Nor will this appear extraordina-
ry, when it is considered that, owing to the application of
telescopes to the purposes of angular measurement, every
imperfection of structure or division becomes magnified
by the whole optical power of that instrument ; and that
thus, not only direct errors of workmanship, arising from
unsteadiness of hand or imperfection of tools, but those
inaccuracies which originate in far more uncontrollable
causes, such as the unequal expansion and contraction of
metallic masses, by a change of temperature, and their
unavoidable flexure or bending by their owai weight, be-
come perceptible and measurable. An angle of one mi-
nute occupies, on the circumference of a circle of 10
inches in radius, only about 3^0^^^ P^^"^ ^^ ^'^ inch, a quan-
tity too small to be certainly dealt wdth wdthout the use
of magnifying glasses ; yet one minute is a gross quan-
tity in the astronomical measurement of an angle. With
the instruments now employed in observatories, a single
second, or the 60th part of a minute, is rendered a dis-
tinctly visible and appreciable quantity. Now, the arc
of a circle, subtended by one second, is less than the
200,000th part of the radius, so that on a circle of 6 feet
in diameter it would occupy no greater linear extent than
j-yL-th part of an inch ; a quantity requiring a poM'erful
microscope to be discerned at all. Let any one figure to
himself, therefore, the difficulty of placing on the circum-
ference of a metallic circle of such dimensions (supposing
the difficulty of its construction surmounted) 360 marks,
dots, or cognizable divisions, which shall be true to their
places within such minute limits ; to say nothing of the
subdivision of the degrees so marked off into minutes, and
of these again into seconds. Such a work has probably
baffled, and will probably for ever continue to baffle, the
utmost stretch of human skill and industry ; nor, if exe-
cuted, could it endure. The ever varying fluctuations of
heat and cold have a tendency to produce not merely tern-

68 A TREATISE ON ASTRONOMY. [cHAP. II.

porary and transient, but permanent, uncompensated
changes of form in all considerable masses of ihose metals
which alone are applicable to such uses ; and their own
weight, however symmetrically formed, must always be
unequally sustained, since it is impossible to apply the
sustaining power to every part separately ; even could
this be done, at all events force must be used to move and
to fix them ; which can never be done without producing
temporary and risking permanent change of form. It is
true, by dividing them on their centres, and in the identi-
cal places they are destined to occupy, and by a thousand
ingenious and delicate contrivances, wonders have been
accomplished in this department of art, and a degree of
perfection has been given, not merely to chefs d'ceuvre,
but to instruments of moderate prices and dimensions, and
in ordinary use, which, on due consideration, must ap-
pear very surprising. But tliough we are entitled to look
for wonders at the hands of scientific artists, we are not
to expect miracles. The demands of the astronomer
will always surpass the power of the artist ; and it must,
therefore, be constantly the aim of the former to make
himself, as far as possible, independent of the imperfec-
tions incident to every work the latter can place in his
hands. He must, therefore, endeavour so to combine his
observations, so to choose his opportunities, and so to
familiarize himself with all the causes which may pro-
duce instrumental derangement, and with all the pecu-
liarities of structure and material of each instrument he
possesses, as not to allow himself to be misled by their
errors, but to extract from their indications, as far as possi-
ble, all that is true, and reject all that is erroneous. It
is in this that the art of the practical astronomer consists,
--^an art of itself of a curious and intricate nature, and of
which we can here only notice some of the leading and
general features.

(104.) The great aim of the practical astronomer be-
ing numerical correctness in the results of instrumental
rneasurement, his constant care and vigilance must be
directed to the detection and compensation of errors,
either by annihilating, or by taking account of, and aU
'pwing for them. Now, if we examine the sources from

CHAP. 11.^ CLASSiriCATION OF SOTTRCES OF KRROR. 09

which errors mar arise in any iiistriimontal determina-
tion, we shall find them chiefly reducible to three prin-
cipal heads : —

(105.) 1st, External or incidental causes of error;
comprehending such as depend on external, uncontrol-
lable circumstances : such as, fluctuations of weather,
which disturb the amount of refraction from its tabu-
lated value, and, being reducible to no fixed law, induce
uncertainty to the extent of their own possible magni-
tude ; such as, by varying the temperature of tlie air,
vary also the form and position of the instruments used,
by altering relative magnitude and the tension of their
parts ; and others of the like nature.

(106.) 2dly, Errors of observation : such as arise, for
example, from inexpertness, defective vision, slowness
in seizing the exact instant of occurrence of a pheno-
menon, or precipitancy in anticipating it, &c. ; from at-
mospheric indistinctness ; insufficient optical power in
the instrument, and the like. Under this head may also
be classed all errors arising from momentary instrumental
derangement, — slips in clamping, looseness of screws, &c.

(107.) 3dly, The third, and by far the most numerous
class of errors to which astronomical measurements are
liable, arise from causes which may be deemed instru-
mental, and which may be subdivided into two principal
classes. The ^rs^ comprehends those which arise from
an instrument not being what it professes to be, which
is error of workmanship. Thus, if a pivot or axis, in-
stead of being, as it ought, exact cylindrical, be slightly
flattened, or elliptical, — if it be not exactly (as it is in-
tended it should) concentric with the circle it carries ; —
if this circle (so called) be in reality 7iot exactly circular,
or not in one plane ; — if its divisions, intended to be
precisely equidistant, should be placed in reality at un-
equal intervals, — and a hundred other things of the same
sort. These are not mere speculative sources of error,
but practical annoyances, Avhich every observer has to
contend with.

C108.) The other subdivision of instrumental errors
comprehends .such as arise from an instrument not being
placed in the posifio7i it ouglit to have ; and from those

70 A TREATISE ON ASTRONOMY, [cHAP. II.

of its parts, whicli are made purposely moveable, not
being properly disposed inter se. These are errors of
adjustment. Some are unavoidable, as they arise from
a general unsteadiness of the soil or building in which
the instruments are placed ; which, though too minute
to be noticed in any other way, become appreciable in
delicate astronomical observations : others, again, are
consequences of imperfect Avorkmanship, as where an
instrument once well adjusted will not remain so, but
keeps deviating and shifting. But the most important
of this class of errors arise from the non-existence of
natural indications, other than those afforded by astrono-
mical observations themselves, whether an instrument
has or has not the exact position, with respect to the
horizon and its cardinal points, the axis of the earth,, or
to other principal astronomical lines and circles, which
it ought to have to fulfil properly its objects.

(109.) Now, with respect to the first two classes of
error, it must be observed, that, in so far as they cannot
be reduced to known laws, and thereby become subjects
of calculation and due allowance, they actually vitiate, to
their full extent, the results of any observations in which
they subsist. Being, however, in their nature casual
and accidental, their effects necessarily lie sometimes
one way, sometimes the other ; sometimes diminishing,
sometimes tending to increase the results. Hence, by
greatly multiplying observations, under varied circum-
stances, and taking the mean or average of their results
this class of errors may be so far subdiied, by setting
them to destroy one another, as no longer sensibly to
vitiate any theoretical or practical conclusion. This is
the great and indeed only resource against such errors not
merely to the astronomer, but to the investigator of nu'
pierical results in every department of physical i^esearch,

{110.) With regard to errors of adjustment and work-
manship, not only the possibility, but the certainty, of
their existence, in every imaginable form, in all instru-
ments, must be contemplated. Human hands or m.ar
chines never formed a circle, drew a straight line, or
erected a perpendicular, nor ever placed an instrument
in perfect adjustment, unless accidentally ; and then only

CttAI>. II.] MUTUAL DESTRUCTION OF ERRORS. 71

during an instant of time. This does not prevent, how-
ever, that a great approximation to all these desiderata
should be attained. But it is the peculiarity of astrono-
mical observation to be the ultimate means of detection
of all mechanical defects which elude by their minute-
ness every other mode of detection. What the eye can-
not discern, nor the touch perceive, a course of astrono-
mical observations will make distinctly evident. The
imperfect products of man's hands are here tested by
being brought into comparison with the perfect work-
manship of nature ; and there is none which will bear
the trial. Now, it may seem like arguing in a vicious
circle, to deduce theoretical conclusions and laws from
observation, and then to turn round upon the instruments
with which those observations were made, accuse them
of imperfection, and attempt to detect and rectify their
errors by means of the very laws and theories which
they have helped us to a knowledge of. A little consi-
deration, however, will suffice to show that such a course
of proceeding is perfectly legitimate.

(111.) The steps by which we arrive at the laws of
natural phenomena, and especially those which depend
for their verification on numerical determinations, are
necessarily successive. Gross results and palpable laAvs
are arrived at by rude observation with coarse instru-
ments, or without any instruments at all ; and these are
corrected and refined upon by nicer scrutiny with more
delicate means. In the progress of this, subordinate
laws are brought into view, which modify both the verbal
statement and numerical results of those which first of-
fered themselves to our notice ; and when these are traced
out, and reduced to certainty, others, again, subordinate
to them, make their appearance, and become subjects of
further inquiry. Now, it invariably happens (and the
reason is evident) that the first glimpse we catch of such
subordinate laws — the first form in which they are
dimly shadowed out to our minds — is that of errors.
We perceive a discordance between what we expect
and what we find. The first occurrence of such a dis-
cordance we attribute to accident. It happens again and
again ; and we begin to suspect our instruments. We

72 A TREATISE ON ASTRONOJttV. [cHAP. II.

then inquire, to what amount of error their determina-
tions can, by possibility, be liable. If their limit of pos-
sible error exceed the observed deviation, we at once
condemn the instrument, and set about improving its
construction or adjustments. Still the same deviations
occur, and, so far from being palliated, are more marked
and better defined than before. We are now sure that
we are on the traces of a law of nature, and we pursue
it till Ave have reduced it to a definite statement, and
verified it by repeated observation, under every variety
of circumstances.

(112.) Now, in the course of this inquiry, it will
not fail to happen that other discordances will strike us.
Taught by experience, we suspect the existence of some
natural law, befoi'e unknown ; we tabulate (i. e. draw out
in order) the results of our observations ; and we per-
ceive, in this synoptic statement of them, distinct indi-
cations of a regular progression. Again Ave improve or
vary our instruments, and we now lose sight of this sup-
posed new law of nature altogether, or find it replaced
by some other, of a totally different character. Thus
we are led to suspect an instrumental cause for what
we have noticed. We examine, therefore, the theory
of our instrument ; Ave suppose defects in its struc-
ture, and, by the aid of geometry, we trace their in-
fluence in introducing actwd errors into its indications.
These errors have their laivs, Avhich, so long as we
have no knowledge of causes to guide us, may be con-
founded with laAvs of nature, and are mixed up Avith
them in their effects. They are not fortuitous, like
errors of observation, but, as they arise from sources
inherent in the instrument, and unchangeable Avhile it
and its adjustments remain unchanged, they are reduci-
ble to fixed and ascertainable forms ; each particular
defect, Avhether of structure or adjustment, producing its
own appropi'iate form of error. When these are tho-
roughly investigated, we recognise among them one
which coincides in its nature and progression with that
of our observed discordances. The mystery is at once
solved: Ave have detected, by direct observation, an in-
strumental defect.

CHAP. II.] DETECTION OF INSTRUMENTAL ERRORS. 73

(113.) It is, therefore, a chief requisite for the practi-
cal astronomer to make himself completely familiar with
the theory of his instruments, so as to be able at once to
decide what effect on his observations any given imperfec-
tion of structure or adjustment will produce in any given
circumstances under which an observation can be made.
Suppose, for example, that the principle of an instrument
required that a circle should be exactly concentric with
the axis on which it is made to turn. As this is a condi-
tion which no workmanship can fulfil, it becomes neces-
sary to inquire what errors will be produced in observa-
tions made and registered on the faith of such an instru-
ment, by any assigned deviation in this respect ; that is
to say, what would be the disagi'cement between obser-
vations made with it and with one absolutely perfect,
could such be obtained. Now, a simple theorem in geo-
metry shows that, whatever be the extent of this devia-
tion, it may be annihilated in its effect on the result of
observations depending on the graduation of the limb,
by the very easy method of reading off the divisions on
two diametrically opposite points of the circle, and tak-
ing a mean ; for the effect of eccentricity is always to
increase one such reading by just the same quantity by
which it diminishes the other. Again, suppose that the
proper use of the instrument required that this axis should
be exactly parallel to that of the earth. As it never can
be placed or remain so, it becomes a question, what
amount of error will arise in its use from any assigned
deviation, whether in a horizontal or vertical plane, from
this precise position. Such inquiries constitute the theory
of instrumental errors ; a theory of the utmost import-
ance to practice, and one of which a complete knowledge
will enable an observer, with very moderate instrumental
means, to attain a degree of precision which might seem
to belong only to the most refined and costly. In the
present work, however, we have no further concern with
it. The few astronomical instruments we propose to de-
scribe in this chapter will be considered as perfect both in
construction and adjustment.

(114.) As the above remarks are very essential to a
right understanding of the philosophy of our subject and

G

74 A TREATISE ON ASTRONOMY. [cHAP. 11.

the spirit of astronomical methods, we shall elucidate
them by taking a case. Observant persons, before the
invention of astronomical instruments, had already con-
cluded the apparent diurnal motions of tlie stars to be
performed in circles about fixed poles in the heavens, as
shown in the foregoing chapter. In drawing this con-
clusion, however, refraction was entirely overlooked, or,
if forced on their notice by its great magnitude in the
immediate neighbourhood of the horizon, was regarded
as a local irregularity, and, as such, neglected or slurred
over. As soon, however, as the diurnal paths of the stars
were attempted to be traced by instruments, even of the
coarsest kind, it became evident that the notion of exact
circles described about one and the same pole would not
represent the phenomena correctly, but that, owing to
some cause or other, the apparent diurnal orbit of every
star is distorted from a circular into an oval form, its
lower segment being flatter than its upper ; and the de-
viation being greater the nearer the star approached the
horizon, the effect being the same as if the circle had
been squeezed upwards from below, and the lower parts
more than the higher. For such an effect, as it was soon
found to arise from no casual or instrumental cause, it
became necessary to seek a natural one ; and refraction
readily occurred to solve the difficulty. In fact, it is a
case precisely analagous to what we have already (art.
47) noticed, of the apparent distortion of the sun near
the horizon, only on a larger scale, and traced up to greater
altitudes. This new law once established, it became ne-
cessary to modify the expression of that anciently re-
ceived, by inserting in it a salvo for the effect of refraction,
or by making a distinction between the apparent diurnal
orbits, as affected by refraction, and the fri<e ones cleared
of that effect.

(115.) Again: The first impression produced by a
view of the diurnal movement of the heavens is, that all
the heavenly bodies perform this revolution in one com-
mon period, viz. a day, or 24 hours. But no sooner do
we come to examine the matter instrument ally, i. e. by
no ling, by timekeepers, their successive arrivals on the
ijiitji'tdian, than we find differences which cannot be ac-

CHAP. IT.] LAWS TRACED BV OBSERVATION. 75

counted for 1)y any error of obsorvution. All the stars,
it is true, occupy the same interval of time between their
successive appulses to the meridian, or to any vertical
circle ; but tliis is a very different one from that occupied
by the sun. It is palpably shorter : being, in fact, only
SG*" 53' 4-09", instead of 24 hours, such hours as our
common clocks mark. Here, then, we have already two
dijft^rent days, a sidereal and a solar ; and if, instead of
the sun, we ol)serve the moon, we find a third, much
longer tlian either, — a lunar day, whose average dura-
tion is 24*' 54™ of our ordinary time, which last is solar
time, being of necessity conformable to the sun's succes-
sive reappearances, on which all the business of life de-
pends.

(116.) Now, all the stars are found to be unanimous
in giving the same exact duration of 23'' 56' 4"-09, for
the sidereal day ; which, therefore, we cannot hesitate to
receive as the period in which the earth makes one revo-
lution on its axis. We are, therefore, compelled to look
on the sun and moon as exceptions to the general law ;
as having a different nature, or at least a different relation
to us, from the stars ; and as having motions, real or ap-
parent, of their own, independent of the rotation of tlie
earth on its axis. Thus a great and most important dis-
tinction is disclosed to us.

(117.) To establish these facts, almost no apparatus is
required. An observer need only station himself to the
north of some well defined vertical object, as the angle
of a building, and placing his eyes exactly at a certain
fixed point (such as a small hole in a plate of metal nail-
ed to some immoveable support), notice the sviccessive
disappearances of any star behind the building, by a
watch.* When he observes the sun, he must shade his
eye with a dark-coloured or smoked glass, and notice the
moments when its western and eastern edges successively

* This is an excellent practical method of asrcrtaining the rate of a
clock or watch, being exceedingly accurate if a few precautions are at-
tended to ; the chief of which is, to take care that that part of the edge
behind which the star (a bright one, not a planet) disappears shall be-
quite smooth; as otherwise variable rcdectioii may transfer the point of
disappearance from a protuberance to a notch, and thus vary the moment
of observation unduly : this is easily secured, by nailing up a smooth
edged board.

78 A TREATISE ON ASTRONOMY. [cHAP. II.

come up to the wall, from which, by taking half the in-
terval he will ascertain (what he cannot directly observe)
the moment of disappearance of its centre.

(118.) When, in pursuing and establishing this gene-
ral fact, we are led to attend more nicely to the times of
the daily arrival of the sun on the meridian, irregulari-
ties (so they first seem) begin to be observed. The inter-
vals between two successive arrivals are not the same at
all times of the year. They are sometimes greater,
sometimes less, than 24 hours, as shown by the clock ;
that is to say, the solar day is not always of the same
length. About the 22st of December, for example, it is
half a minute longer, and about the same day of Septem-
ber nearly as much shorter, than its average duration.
And thus a distinction is again pressed upon our notice
between the actual solar day, which is never two days in
succession alike ; and the mean solar day of 24 hours,
which is an average of all the solar days throughout the
year. Here, then, a new source of inquiry opens upon
us. The sun's apparent motion is not only not the same
Avith that of the stars, but it is not (as the latter is) uni-
form. It is subject to fluctuations, whose laws become
matter of investigation. But to pursue these laws, we
require nicer means of observation than what we have
described, and are obliged to call into our aid an insti-u-
ment called the transit instrument, especially destined
for such observations, and to attend minutely to ad the
causes of irregularity in the going of clocks and watches
which may affect our reckoning of time. Thus we be-
come involved by degrees in more and more delicate in-
strumental inquiries ; and we speedily find that, in pro-
portion as we ascertain the amount and law of one great
or leading fluctuation, or inequality, as it is called, of the
sun's diurnal motion, we bring into view others continu-
ally smaller and smaller, Avhich were before obscured, or
mixed up with errors of observation and instrumental im-
perfections. In short, we may not inaptly compare the
mean length of the solar day to the mean or average
height of water in a harbour, or the general level of the
pea unagitated by tide or waves. The great annual fluc-
tuation above noticed may be compared to the daily varia-

CHAP. 11.3 OF TIME AND ITS MEASUREMENT. 77

ions of level produced by the tides, which are nothing
but enormous waves extending- over the whole ocean,
while the smaller subordinate inequalities may be assi-
milated to waves ordinarily so called, on which, when
large, we perceive lesser undulations to ride, and on these
again, minuter rinplings, to tlie series of whose subordi-
nation we can perceive no end.

(119.) With the causes of these irregulainties in the
solar motion we have no concern at present ; their expla-
nation belongs to a more advanced part of our subject;
but the distinction between tlie solar and sidereal days, as
it pervades every part of astronomy, requires to be early
introduced, and never lost sight of. It is, as already ob-
served, the mean or average length of the solar day,
which is used in the civil reckoning of time. It com-
mences at midnight, but astronomers (at least those of
this country), even when they use mean solar time, de-
part from the civil reckoning, commencing their day at
noon, and reckoning the hours from 0 round to 24.
Thus, 1 1 o'clock in the forenoon of the second of Janu-
ary, in the civil reckoning of time, corresponds to January
1 day 23 hours in the astronomical reckoning ; and one
o'clock in the afternoon of the former, to January 2 days
1 hour of the latter reckoning. This usage has its ad-
vantages and disadvantages, but the latter seem to pre-
ponderate ; and it would be well if, -in consequence, it
could be broken through, and the civil reckoning substi-
tuted.

(120.) Both astronomers and civilians, however, who
inhabit different points of the earth's surface, differ from
each other in their reckoning of time ; as it is obvious
they must, if we consider that, when it is noon at one
place, it is midnight at a place diametrically opposite ;
sunrise at another ; and sunset, again, at a fourth. Hence
arises consid^erable inconvenience, especially as respects
places differing very widely in situation, and which may
even in some critical cases involve the mistake of a whole
day. To obviate this inconvenience, there has lately
been introduced a system of reckoning time by mean so-
lar days and parts of a day counted from a fixed instant,
common to all the world, and determined by no local cir-

g2

78 A TREATISE ON ^ASTRONOMY. [cHAP. II.

cumstance, such as noon or midnight, but by the motion
of the sun among the stars. Time, so reckoned, is called
equinoctial time, and is numerically the same, at the same
instant, in every part of the globe. Its origin will be ex-
plained more fully at a moi'e advanced stage of our work.

(121.) Time is an essential element in astronomical
observation, in a twofold point of view: — 1st, As the
representative of angular motion. The earth's diurnal
motion being uniform, every star describes its diurnal cir-
cle uniformly ; and the time elapsing between the pas-
sage of the stars in succession across the meridian of any
observer becomes, therefore, a direct measure of their dif-
ferences of right ascension. 2dly, As the fundamental
element (or, independent variable, to use the language of
geometers) in all dynamical theories. The great object of
astronomy is the determination of the laws of the celestial
motions, and their reference to their proximate or remote
causes. Now, the statement of the Imv of any observed
motion in a celestial object can be no other than a propo-
sition declaring what has been, is, and will be, the real
or apparent situation of that object at any time past, pre-
sent, or future. To compare such laws, therefore, with
observation, we must possess a register of the observed
situations of the object in question, and of the times ivhen
they were observed.

(122.) The measurement of time is performed by
clocks, chronometers, clepsydras, and hour-glasses : the
two former are alone used in modern astronomy. The
hour-glass is a coarse and rude contrivance for measuring,
or rather counting out, fixed portions of time, and is en-
tirely disused. The clepsydra, which measured time by
the gradual emptying of a large vessel of water through a
determinate orifice, is susceptible of considerable exact-
ness, and was the only dependence of astronomers before
the invention of clocks and watches At present it is
abandoned, owing to the greater convenience and exact-
ness of the latter instruments. In one case only has the
revival of its use been proposed ; viz. for the accurate
measurement of very small portions of time, by the flow-
ing out of mercury from a small orifice in the bottom of
a vessel, kept constantly full to a fixed height. The

CHAP. 11.3 CLOCKS CHRONOMETERS. 79

Stream is intercepted at the moment of noting any event,
and directed aside into a receiver, into which it continues
to run, till the moment of noting any other event, when
the intercepting cause is suddenly removed, the stream
flows in its original course, and ceases to run into the
receiver. The weight of mercury received, compared
with the weight received in an interval of time observed
by the clock, gives the interval between the events ob-
served. This ingenious and simple method of resolving,
with all possible precision, a problem which has of late
been much agitated, is due to Captain Kater.

(123.) The pendulum clock, however, and the balance
watch, with those improvements and refinements in its
structure which constitute it emphatically a chronometer,*
are the instruments on which the asti'onomer depends
for his knowledge of the lapse of time. These instru-
ments are now brought to such perfection, that an irregu-
larity in the rate of going, to the extent of a single se-
cond in twenty-four hours in two consecutive days, is not
tolerated in one of good character; so that any interval
of time less than twenty-four hours may be certainly
ascertained within a ievf tenths of a second, by their use.
In proportion as intervals are longer, the risk of error, as
well as the amount of error risked, becomes greater, be-
cause the ai'cidental errors of many days may accumu-
late ; and causes producing a sIoav progressive change in
the rate of going may subsist unperceived. It is not safe,
therefore, to trust the determination of time to clocks, or
watches, for many days in succession, without checking
them, and ascertaining their errors by reference to natu-
ral events which we know to happen, day after day, at
equal intervals. But if this be done, the longest intervals
maybe fixed with the same precision as the shortest;
since, in fact, it is then only the times intervening be-
tween the first and last moments of such long intervals,
and such of those periodically recurring events adopted
for our points of reckoning, as occur within twenty-four
hours respectively of either, that we measure by artifi-
cial means. The whole days are counted out for us by
nature ; the fractional parts only, at either end, are mea-
* x««>">f, time ; i"«Tt£i»,to measure.

80 A TREATISE ON ASTRONOMY. [cHAP. II.

sured by our clocks. To keep the reckoning of the inte-
ger days correct, so that none shall be lost or counted
twice, is the object of the calendar. Chronology marks
out the order of succession of events, and refers them to
their proper years and days ; while chronometry, ground-
ing its determinations on the precise observation of such
regularly periodical events as can be conveniently and
exactly subdivided, enables us to fix the moments in
which phenomena occur, with the last degree of preci-
sion.

(124.) In the culmination, or transit (i. e. the pas-
sage across the meridian of an observer) of every star in
the heavens, he is furnished with such a regularly pe-
riodical natural event as we allude to. Accordingly, it is
to the transits of the brightest and most conveniently
situated fixed stars that astronomers resort to ascertain
their exact time, or, which comes to the same thing, to
determine the exact amount of error of their clocks.

(125.) The instrum.ent with wliich the culminations of
celestial objects are observed is called a transit instru-
ment. It consists of a telescope firmly fastened on a hori-
zontal axis directed to the east and wes-t points of the
horizon, or at right angles to the plane of the meridian of
the place of observation. The extremities of the axis
are formed into cylindrical pivots of exactly equal diame-
ters, which rest in notches formed in metallic supports,
bedded (in the case of large instruments) on strong piers
of stone, and susceptible of nice adjustment by screws,
both in a vertical and horizontal direction. By the for-

mer adjustment, the axis can be rendered precisely hori-l'"'
zontal, by levelling it with a level made to rest on the

CHAP. 11. J TRANSIT INSTRUMENT. 81

pivots. By the latter adjustment the axis is brought pre-
cisely into the east and west direction, the criterion of
which is furnished by the observations themselves made
with the instrument, or by a well-defined object called a
meridian mark, originally determined by such observa-
tions, and then, for convenience of ready reference, per-
manently established, at a great distance, exactly in a
meridian line passing through the central point of the
whole instrument. It is evident, from this dcpcription,
that, if the central line of the telescope (iliat wliich joins
the centres of its object-glass and eye-glass, and which
is called in astronomy its line of coUimalion) be once well
adjusted at right angles to the axis of the ti-ans.ii, it will
never quit the plane of the meridian, when the instrument
is turned round on its axis.

(126.) In the focus of the eye-piece, and at right an-
gles to the length of the telescope, is placed a system of
one horizontal and five equidistant vertical threads or
wires, as represented in the annexed figure, which always
appear in the field of view, when properly illuminated,

by day by the light of the sky, by night by that of a lamp
introduced by a contrivance not necessary here to explain.
The place of this system of wires may be altered by ad-
justing screws, giving it a lateral (horizontal) motion ; and
it is by this means brought to sitch a position, that the
middle one of the vertical wires shall intersect the line of
collimation of the telescope, where it is arrested and
permanently fastened. In this situation it is evident
that the middle thread will be a visible representation of
that portion of the celestial meridian to wliich the tele-
scope is pointed ; and when a star is seen to cross this
wire in the telescope, it is in the act of culminating, or
passing the celestial meridian. The instant of this event is

82 A TREATISE ON ASTRONOMY. [cHAP. II.

noted by the clock or chroaomeler, which forms an in-
dispensable accompaniment of the transit instrument.
For greater precision, tlie moments of its crossing all tlie
five vertical threads is noted, and a mean taken, which
(since the threads are equidistant) would give exactly the
same result, were all the observations perfect, and will,
of course, tend to subdivide and destroy their errors in
an average of the whole.

(127.) For the mode of executing the adjustments,
and allowing for the errors unavoidable in the use of this
simple and elegant instrument, the reader must consult
works especially devoted to this department of practical
astronomy.* We shall here only mention one import-
ant verification of its correctness, which consists in re-
versing the ends of the axis, or turning it east for west.
If this be done, and it continue to give the same results,
and intersect the same point on the meridian mark, Ave
may be sure that the line of collimation of the telescope
is truly at right angles to the axis, and describes strictly
a plane, i. e. marks out in the heavens a great circle. In
good transit observations, an error of two or three tenths
of a second of time in the moment of a star's culmination
is the utmost which need be apprehended, exclusive of
the error of the clock : in other words, a clock may be
compared witli the earth's diurnal motion by a single
observation, without risk of greater error. By multiply-
ing observations, of course, a yet greater degree of pre-
cision may be obtained.

(128.) The angular intervals measured by means of
tlie transit instrument and clock are arcs of the equinoc-
tial, intercepted between circles of declination passing
through the objects observed ; and their measurement,
in this case, is performed by no artificial graduation of
circles, but by the help of the earth's diurnal motion,
which carries equal arcs of the equinoctial across the
meridian, in equal times, at the rate of 15° per sidereal
hour. In all other cases, when we would measure an-
gular intervals, it is necessary to have recourse to cir-
cles, or portions of circles, constructed of metal or other

* See Dr. Pearson's Treatise on Practical Astronomy. Also Bianchi
Sopra lo Stromenio de' Passagi. Ephein- di Milano, 1824,

CHAP. II.] MEASUREMENT OF ANGLES. 83

firm and tlurable material, and mechanically subdivided
into equal parts, such as degrees, minutes, &c. Let
A BOD be such a circle, divided into 360 degrees (niun-

S

bered in order from any point 0° in the circumference,
round to the same point again), and connected with its
centre by spokes or rays, xyz, firmly united to its cir-
cumference or limb. At the centre let a circular hole be
pierced, in which shall move a pivot exactly fitting it,
carrying a tube, whose axis, ab, is exactly parallel to
the plane of the circle, or perpendicular to the pivot ; and
also the two arms m n, at right angles to it, and forming
one piece with the tube and the axis ; so that the motion
of the axis on the centre shall carry the tube and arms
smoothly round the circle, to be arrested and fixed at any
point we please, by a contrivance called a clamp. Sup-
pose, now, we would measure the angular interval be-
tween two fixed objects, ST. The plane of the circle
must first be adjusted so as to pass through them both.
This done, let the avis « 6 of the tube be directed to
one of them, S, and clamped. Then will a mark on the
arm m point either exactly to some one of the divisions
on the limb, or between two of them adjacent. In the
former case, the division must be noted, as the reading
of the arm m. In the latter, the fractional part of one
whole interval between the consecutive divisions by
which the mark on m surpasses the last inferior division
must be estimated or measured by some mechanical or
optical means. (See art. 130.) The division and frac-
tional part thus noted, and reduced into degrees, minutes,
and seconds, is to be set down us the reading of the limb

84 A TREATISE ON ASTRONOMY. [cHAP. 11.

corresponding to that position of the tube ab, where it
points to the object S. Tlie same must then be done for
the object T ; the tube pointed to it, and the limb " read
off.''"' It is manifest, then, that, if the lesser of these
readings be subtracted from the greater, tlidr difference
will be the angular interval between S and T, as seen
from the centre of the circle, at whatever point of the
limb the commencement of the graduations on the point
0° be situated.

(129.) The very same result will be obtained, if, in-
stead of making the tube moveable upon the circle, we
connect it invariably with the latter, and make both re-
volve together on an axis concentric Avith the circle, and
forming one piece with it, working in a hollow formed
to receive and fit it in some fixed support. Such a com-
bination is represented in section in the annexed sketch.
T is the tube or sight, fastened, at pp, on the circle AB,

J
f.

whose axis, D, works in the solid metallic centring E,
from Avhich originates an arm, F, carrying at its ex-
tremity an index, or other proper mark, to point out and
read ofl' the exact division of the circle at B, the point
close to it. It is evident that, as the telescope and circle
revolve through any angle, the part of the limb of the
latter, which by such revolution is carried past the index
F, will measure the angle described. This is the most
usual mode of applying divided circles in astronomy.

(130.) The index F may either be a simple pointer,
like a clock hand (fig- ct) ; or a vernier (fig. b) ; or,

/^

CHAP. II.]] APPLICATION OF THE TELESCOPE. 66

lastly, a compound microscope (fig. c), represented in
section (in fig. d), and furnished with a cross in the
common focus, of its ohject and eye-glass, moveable by
a fine threaded screw, by which the intersection of the
cross may be brought to exact coincidence with the
image of the nearest of the divisions of the circle ; and by
the turns and parts of a turn of the screw required for this
purpose the distance of that division from the original
or zero point of the microscope may be estimated. This
simple but delicate contrivance gives to the reading off
of a circle a degree of accuracy only limited by the power
of the microscope, and the perfection with which a screw
can be executed, and places the subdivision of angles on
the same footing of optical certainty which is introduced
into their measurement by the use of the telescope.

(131.) The exactness of the result thus obtained must
depend, 1st, on the precision with which the tube a b
can be pointed to the objects ; 2dly, on the accuracy of
graduation of the limb ; 3dly, on the accuracy with
which the subdivision of the intervals between any two
consecutive graduations can be accomplished. The
mod-e of accomplishing the latter object with any re-
quired exactness has been explained in the last article.
With regard to the graduation of the limb, being merely
of a mechanical nature, we shall pass it Avithout remark,
further than this, that, in the present state of instrument
making, the amount of error from this source of inaccu-
racy is reduced within very narrow limits indeed. With
regard to the first, it must be obvious that, if the sights
a 6 be nothing more than what they are represented in
the figure (art. 128), simple crosses or pin-holes at the
ends of a hollow tube, or an eye-hole at one end, and a
cross at the other, no greater nicety in pointing can be
expected than what simple vision with the naked eye
can command. But if, in place of these simple but
coarse contrivances, the tube itself be converted into a
telescope, having an object-glass at b, and an eye-piece
at a ; and if the motion of the tube on the limb of the
circle be arrested when the object is brought just into
the centre of the field of view, it is evident that a greater
degree of exactness may be attained in the pointing of

H

86 A TREATISE ON ASTRONOMY. [cHAP. th

the tube tlian by the unassisted eye, in proportion to the
magnifying power and distinctness of the telescope used.
The last attainable degree of exactness is secured by
stretching in the common focus of the object and eye-
glasses two delicate fibres, such as fine hairs or spider-
lines, intersecting each other at right angles in the centre
of the field of view. Their points of intersection afford
a permanent mark with which the image of the object
can be brought to exact coincidence by a proper degree
of caution (aided by mechanical contrivances), in bringing
the telescope to its final situation on the limb of the circle,
and retaining it there till the "reading off" is finished.
(132.) This application of the telescope may be con-
sidered as completely annihilating that part of the error
of observation which might otherwise arise from errone-
ous estimation of the direction in Avhich an object lies
from the observer's eye, or from the centre of the in-
strument. It is, in fact, the grand source of all the pre-
cision of modern astronomy, without which all other re-
finements in instrumental Avorkmanship would be thrown
away; the errors capable of being committed in point-
ing to an object, without such assistance, being far greater
than what could arise from any but the very coarsest
graduation.* In fact, the telescope thus applied becomes,

* The honour of this capital improvement has been successfully vin-
dicated by Derham (Phil. Trans, xxx. 603) to our young, talented, and
unfortunate countryman Gascoigne, from his correspondence with Crab-
tree and Horrockes, in his (Derham's) possession. The pa.ssages cited
by Derham from these letters leave no doubt that, so early as 1640,
Gascoigne had applied telescopes to his quadrants and sextants, with
threads in the common focus of the glasses ; and had even carried the in-
vention so far as to illuminate the field of view by artificial light, -which
he found " very helpful when the moon appeareth not, or it is not otherwise
light enough." These inventions were freely communicated by him to
Crabtree, and through him to his friend Horrockes, the pride and boast
of British astronomy ; both of whom expressed their unbounded admira-
tion of this and many other of his delicate and admirable improvements
in the art of observation. Gascoigne, however, perished at the age of
twenty-three at the battle of Marston Moor ; and the premature and
sudden death of Horrockes, at a yet earlier age, will account for the
temporary oblivion of the invention. It was revived, or re-invented, in
1667, by Picard and Auzout (Lalande, Astron. 2310), after which its use
became universal. Morin, even earlier than Gascoigne (in 1635), had
proposed to substitute the telescope for plain sights ; but it is the thread
or wire stretched in the focus with which the image of a star can be
brought to exact coincidence, which gives the telescope its advantage in
practice ; and the idea of this does not seem to have occurred to Morin.
(3ee Lalande, ttbi supra.)

CHAP. II.] INTERVALS IN DECLINATION MEASURED. 87

with respect to angular, what the microscope is with
respect to linear dimension. By concentrating attention
on its smallest points, and magnifying into palpable in-
tervals the minutest differences, it enables us not only to
scrutinize the form and structure of the objects to which
it is pointed, but to refer their apparent places, Avith all
but geometrical precision, to the parts of any scale with
which we propose to compare them.

(133.) The simplest mode in which the measurement
of an angular interval can be executed, is what we have
just described ; but, in strictness, this mode is applicable
only to terrestrial angles, such as those occupied on the
sensible horizon by the objects which surround our sta-
tion,— because these only remain stationary during the
interval while the telescope is shifted on the limb from
one object to the other. But the diurnal motion of tlie
heavens, by destroying this essential condition, renders
the direct measurement of angular distance from object
to object by this means impossible. The same objection,
however, does not apply if we seek only to determine
the interval between the diurnal circles described by any
two celestial objects. Suppose every star, in its diurnal
revolution, were to leave behind it a visible trace in the
heavens, — a fine line of light, for instance, — then a teles-
cope once pointed to a star, so as to have its image
brought to coincidence v/ith the intersection of the wires,
would constantly remain pointed to some portion or other
of this line, which would thercibre continue to appear
in its field as a luminous line, permanently intersecting
the same point, till the star came round again. From
one such line to another the telescope might be shifted,
at leisure, without error ; and then the angular interval
between the two diurnal circles, in the plane of the tele-'
scope's rotation, might be measured. Now, though we
cannot see the path of a star in the heavens, we can wait
till the star itself crosses the field of view, and seize the
moment of its passage to place the intersection of its
wires so that the star shall traverse it ; by which, when
the telescope is well clamped, we equally well secure the
position of its diurnal circle as if we continusd to see it
ever so long. The reading off of the limb may then be

88 A TREATISE ON ASTRONOMY. [CHAP. II.

performed at leisure ; and when another star comes
round into the plane of the circle, we may unclamp the
telescope, and a similar observation will enable us to as-
sign the place of its diurnal circle on the limb : and the
observations may be repeated alternately, every day, as
the stars pass, till Ave are satisfied with their result.

(134.) This is the principle of the mvu-al circle, which
is nothing more than such a circle as we have described
in art. 129, firmly supported, in the plane of the meri-
dian, on a long and powerfvd horizontal axis. This axis
is let into a massive pier, or wall, of stone (whence the
name of the instrument), and so secured by screws as to
be capable of adjustment both in a vertical and horizon-
tal direction ; so that, like the axis of the transit, it can
be maintained in the exact direction of the east and west
points of the horizon, the plane of the circle being con-
sequently truly meridional.

(135.) The meridian, being at right angles to all the
diurnal circles described by the stars, its arc intercepted
between any two of them will measure the least distance
between these circles, and will be equal to the difference
of the declinations, as also to the difference of the meri-
dian altitudes of the objects — at least when corrected
for refraction. These differences, then, are the angular
intervals directly measured by the mural circle. But
from these, supposing the law of refraction known, it is
easy to conclude, not their differences only, but the
quantities themselves, as we shall now explain.

(136.) The declination of a heavenly body is the com-
plement of its distance from the pole. The pole, being
a point in the meridian, might be directly observed on the
limb of the circle, if any star stood exactly therein ; and
thence the polar distances, and of course, the declina-
tions of all the rest, might be at once determined. But
this not being the case, a bright star as near the pole as
can be found is selected, and observed in its upper and
lower culminations ; that is, when it passes the meridian
above and below the pole. Now, as its distance from
the pole remains the same, the difference of reading off
the circle in the two cases is, of coui'se (when connected
for refraction), equal to twice the polar distance of the

MURAL CIRCLE.

89

CHAP. 11.]

star ; the arc intercepted on the limb of the circle being,
in this case, equal to the angular diameter of the star's
diurnal circle. In the annexed diagram, HPO represents
the celestial meridian, P the pole, BR, AQ, CD, the di-
urnal circles of stars which arrive on the meridian — at
BA and C in their upper, and at RQD in their lower cul-

minations, of which D happens above the horizon HO.
P is the pole ; and if we suppose hp o to be the mural
circle, having S for its centre, b a cp d will be the points
on its circumference corresponding to BACPD in the
heavens. Now, the arcs b a, b c,b d, and c d are given
immediately by observation ; and since CP=PD, we
have also cp=p d, and each of them =i'C d, consequently
the place of the polar po'mt, as it is called, upon the limb
of the circle becomes known, and the arcs pb,p a,p c,
which represent on the circle the polar distances re-
quired become also known.

(137.) The situation of the pole star, which is a very
brilliant one, is. eminently favourable for this purpose,
being only about a degi-ee and a half from the pole ; it
is, therefore, the star usually and almost solely diosen
for this important purpose ; the more especially because,
both its culminations taking place at great and not very
different altitudes, the refractions by which they are
affected are of small amount, and differ but slightly from
each other, so that their correction is easily and safely
applied. The brightness of the pole star, too, allows
it to be easily observed in the daytime. In consequence

h9

90 A TREATISE ON ASTRONOMV. [cHAP. II.

of these peculiarities, this star is one of constant resort
with astronomers for the adjustment and verification of
instruments of almost every description. In the case of
the transit, for example, it furnishes a ready means of
ascertaining whether the plane of the telescope's motion
is coincident with the meridian. For since this latter
plane bisects its diurnal circle, the eastern and western
portion of it require equal times for their description.
Let, therefore, the moments of its transit above and be-
low the pole be noted ; and if they are found to follow
at equal intervals of 13 sidereal hours, we may conclude
with certainty that the plane of the telescope's motion is
meridional, or the position of its horizontal axis exactly
east and west. But if it pass from one to the other ap-
parent culmination in unequal intervals of time, it is
equally certain that an extra-meridional error must exist,
the deviation lying towards that side on which the least
interval is occupied. And the axis must be moved in
azimuth accordingly, till the difference in question dis-
appears on repeating the observations.

(138.) The place of the polar point on the limb of
the mural circle once determined, becomes an origin, or
zero point, from which the polar distances of all objects,
referred to other points on the same lines, reckon. It
matters not whether the actual commencement 0° of the
graduations stand there, or not ; since it is only by
the difference of the readings that the arcs on the
limb are determined ; and hence a great advantage is
obtained in the power of commencing anew a fresh series
of observations, in which a different part of the circum-
ference of the circle shall be employed, and different
graduations brought into use, by which inequalities of
division may be detected and neutralized. This is ac-
complished practically by detaching the telescope from
its old bearings on the circle, and fixing it afresh on a
different part of the circumference.

(139.) A point on the limb of the mural circle, not
less important than the polar point, is the horizontal
point, which, being once known, becomes in like man-
ner an origin, or zero point, from which altitudes are
reckoned, The principle of its determination is ulti'

CHAP. II. J POLAR AND HORIZONTAL POINTS. 91

mately nearly the same with tliat of the polar point.
As no star exists in the celestial horizon, the observer
must seek to determine two points on the limb, the one
of which shall be precisely as far below the horizontal
point as the otlier is above it. For this purpose, a star
is observed at its culmination on one night, by pointing
the telescope directly to it, and the next, by pointing to
the image of the same star reflected in the still, unruffled
surface of a fluid at perfect rest. Mercury, as the most
reflective fluid known, is generally chosen for that use.
As the surface of a fluid at rest is necessarily horizontal,
and as the angle of reflection, by the laws of optics, is
equal to that of incidence, this image will be just as
much depressed below the horizon, as the star itself is
elevated above it (allowing for the diff'erence of refrac-
tion at the moments of observation). The arc inter-
cepted on the limb of the circle between the star and its
reflective image thus consecutively observed, when cor-
rected for refraction, is the double altitude of the star,
and its point of bisection the horizontal point. The re-
flecting surface of a fluid so used for the determination
of the altitudes of objects is called an artificial horizon.
(140.) The mural circle is, in fact, at the same time, a
transit instrument ; and, if furnished with a proper sys-
tem of vertical wires in the focus of its telescope, may
be used as such. As the axis, however, is only support-
ed at one end, it has not the strength and permanence ne-
cessary for the more delicate purposes of a transit ; nor
can it be verified, as a transit may, by the reversal of the
two ends of its axis, east for west. Nothing, however,
prevents a divided circle being permanently fastened on
the axis of a transit instrument, near to one of its extre-
mities, so as to revolve with it, the reading off" being per-
formed by a microscope fixed on one of its piers. Such
an instrument is called a transit circle, or a meridian
CIRCLE, and serves for the simultaneous determination of
the right ascensions and polar distances of objects ob-
served with it ; the time of transit being noted by the clock,
and the circle being read off" by the lateral microscope.

(141.) The determination of the horizontal point on
the limb of an instrument is of such essential importance
in astronomy, that the student should be ma4e acquaint-

92 A TREATISE ON ASTRONOTiIY. [[cHAP. II.

ed with every means employed for this purpose. These
are the artificial horizon, the plumb-line, the level, and the
floating collimator. The artificial horizon has been al-
ready explained. The plumb-line is a fine thread or wire,
to which is suspended a weight, whose oscillations are
impeded and quickly reduced to rest by plunging it in
water. The direction ultiirately assumed by such a line,
admitting its perfect fiexibility , is that of gravity, or per-
pendicular to the surfac3 of still water. Its application
to the purposes of astronomy is, however, so delicate, and
difficult, and liable to error, unless extraordinary precau-
tions are taken in its v.S3, that it is at present almost uni-
versally abandoned, for the more convenient and equally
exact instrument the level.

(142.) The level '"I nothing more than a glass tube
nearly filled with a liquid (spirit of wine being that now
generally used, on account of its extreme mobility, and
not being liable to freeze), the bubble in which, when the

tube is placed horizontally, would rest indifferently in any
part if the tube could be mathematically straight. But
that being impossible to execute, and every tube having
some slight curvature, if the convex side be placed up-
wards, the bubble v/i!l cccupy the higher part, as in the
figure (where the curvature is purposely exaggerated).
Suppose such a tube as AB firmly fastened on a straight
bar, CD, and marked at a b, two points distant by the
length of the bubble ; then, if the instrument be so placed
that the bubble shall occupy this interval, it is clear that
CD can have no other than one definite inclination to the
horizon ; because, were it ever so little moved one way
or other, the bubble would shift its place, and run towards
the elevated side. Suppose, now, that we would ascer-
tain whether any given line PQ be horizontal ; let the
base of the level CD be set upon it, and note the points

CHAP. II.] OF THE LEVEL. 93

a b, between Avhich the bubble is exactly contained ; then
turn the level end for end, so that C shall rest on Q, and
D on P. If then the bubble continue to occupy the same
place between a and b, it is evident that PQ can be no
otherwise than horizontal. If not, the side towards which
the bubble runs is highest, and must be lowered. Astro-
nomical levels are furnished with a divided scale, by
which the places of the ends of the bubble can be nicely
marked ; and it is said that they can be executed with
such delicacy, as to indicate a single second of angular
deviation from exact horizontality.

(143.) The mode in which a level may be applied to
find the horizontal point- on the limb of a vertical divided
circle may be thus explained : Let AB be a telescope
firmly fixed to such a circle, DEF, and moveable in one

I

with it on a horizontal axis C, which must be like that of
a transit, susceptible of reversal (see art. 127), and with
which the circle is inseparably connected. Direct the
telescope on some distant well-defined object S, and bi-
sect it by its horizontal Avire, and in this position clamp
it fast. Let L be a level fastened at right angles to an
arm, LEF, furnished with a microscope, or vernier at F,
and, if we please, another at E. Let this arm be fitt2d by
grinding on the axis C, but capable of moving smoothly
on it without carrying it round, and also of being clamped
fast on it, so as to prevent it from moving until required.
While the telescope is kept fixed on the obiect S, let the

94 A TREATISE ON ASTRONOMY. [cHAP. II.

level be set so as to bring its bubble to the marks a b, and
clamp it there. Then will the arm LCF have some cer-
tain determinate inclination (no matter what) to the hori-
zon. In this position let the circle be read oil' at F, and
then let the Avhole apparatus be reversed by turning its
horizontal axis end for end, ivithout imclamping the level
arm from the axis. This done, by the motion of the
whole instrument (level and all) on its axis, restore the
level to its horizontal position with the bubble at a b.
Then we are sure that the telescope has now the same
inclination to the horizon the other ivay, that it had when
pointed to S, and the reading off at F will not have been
changed. Now, unclamp the level, and, keeping it nearly
horizontal, turn round the circle on the axis, so as to car-
ry back the telescope through the zenith to 8, and in
that position clamp the circle and telescope fast. Then it
is evident that an angle equal to twice the zenith distance
of S has been moved over by the axis of the telescope
from its last position. Lastly, without unclamping the
telescope and circle, let the level be ouce more rectified.
Then will the arm LEF once more assume the same de-
finite position with respect to the horizon ; and, conse-
quently, if the circle be again read off, the difference be-
tween this and the previous reading must measure the
arc of its circumference which has passed under the
point F, which may be considered as having all the
while retained an invariable position. This difference,
then, will be the double zenith distance of S, and its half
the zenith distance simply, the complement of which is
its altitude. Thus the altitude corresponding to a given
reading of the limb becomes known, or, in other words,
the horizontal point on the limb is ascertained. Circuit-
ous as this process may appear, there is no other mode
of employing the level for this purpose which does not
in the end come to the same thing. Most commonly,
however, the level is used as a mere fiducial reference,
to preserve a hoi'izontal point once well determined by
other means, which is done by adjusting it so as to stand
level when the telescope is truly horizontal, and thus
leaving it depending on the permanence of its adjustment.
(144.) The last, but probably not the least exact, as it

CHAP, n.] THE FLOATING COLLIMATOR. 95

certainly is, in innumerable cases, the most convenient
means of ascertaining the Iwrizontul point, is that af-
forded by the floating collimator, a recent invention of
Captain Kater. This elegant instrument is nothing more
than a small telescope furnished with a cross-wire in its
focus, and fastened horizontally, or as nearly so as may
be, on a flat iron fioat, which is made to swim on mer-
cury, and which, of course, will, when left to itself, as-
sume always one and the same invariable inclination to
the horizon. If the cross-wires of the collimator be illu-

minated by a lamp, being in the focus of its object-glass,
the rays from them will issue parallel, and will therefore
be in a fit state to be brought to a focus by the object-
glass of any other telescope, in which they will form an
image as if they came from a celestial object in their di-
rection, i. e. at an altitude equal to their inclination.
Thus the intersection of the cross of the collimator may
be observed as if it were a star, and that^ however near
the two telescopes are to each other. By transferring then,
the collimator still floating on a vessel of mercury from
the one side to the other of a circle, we are furnished with,
two quasi-celestial objects, at precisely equal altitudes,
on opposite sides of the centre ; and if these be observed
in succession with the telescope of the circle, bringing its
icross to bisect the image of the cross of the collimator (for
which end the wires of the the latter cross
are purposely set 45° inclined to the hori-
zon) the difference of the readings on its limb
will be twice the zenith distance of either ;
whence, as in the last article, the horizontal
or zenith point is immediately determined.*

* Another, and, in many respects, preferable form of the floating colli-
mator, in whicli the telescope is vertical, and whereby the zenith point is
directly ascertained, is described in the Phil. Trans. 1828, p. 257 bv the
same author.

**'., '\.

\// :

'\/

•-■•■.•-'

y

■*'

■\ »

96 A TREATISE ON ASTRONOMY. []cHAP. 11.

(145.) The transit and mural circle are essentially me-
ridian instruments, being used only to observe the stars
at the moment of their meridian passage. Independent
of this being the most favourable moment for seeing them,
it is that in which their diurnal motion is parallel to the
horizon. It is therefore easier at this time than it could
be at any other, to place the telescope exactly in their
true direction ; since their apparent course in the field of
view being parallel to the horizontal thread of the system
of wires therein, they may, by giving a fine motion to
the telescope, be brought to exact coincidence with it,
and time may be allowed to examine and correct this co-
incidence, if not at first accurately hit, Avhich is the case
in no other situation. Generally speaking, all angular
magnitudes, which it is of importance to ascertain ex-
actly, should, if possible, be observed at their maxima or
minima of increase or diminution; because at these
points they remain not perceptibly changed during a time
long enough to complete, and even, in many cases, to re-
peat and verify our observations in a careful and leisurely
manner. The angle which, in the case before us, is in
this predicament, is the altitude of the star, Avhich attains
its maximum or minimum on the meridian, and which is
measured on the limb of the mural circle.

(146.) The purposes of astronomy, however, require
that an observer should possess the means of observing
any object not directly on the meridian, but at any point
of its diurnal course, or wherever it may present itself
in the heavens. Noav, a point in the sphere is determined
by reference to two great circles at right angles to each
other ; or of two circles one of which passes tlirough the
pole of the other. These, in the language of geometry,
are co-ordinates by which its situation is ascertained :
for instance, — on the earth, a place is known if we know
its longitude and latitude ; — in the starry heavens, if we
know its right ascension and declination ; — in the visible
hemisphere, if we know its azimuth and altitude, &c.

(147.) To observe an object at any point of its diurnal
course, we must possess the means of directing a tele-
scope to it; which, therefore, must be capable of motion
in • two planes at right angles to each other ; and the

CHAP. II.]

CO-ORDINATE CIRCLES.

97

amount of its an^ilar motion in each must be measured
on two circles co-ordinate to each other, \vhose planes
must be parallel to those in which the telescope movies.
The practical accomplishment of this condition is effect-
ed by making the axis of one of the circles penetrate that
of the other at right angles. The pierced axis turns on
fixed supports, while the other has no connexion with
any external support, but is sustained entirely by that
which it penetrates, which is strengthened and enlarged
at the point of penetration to receive it. The annexed
figure exhibits the simplest form of such a combination,
though by no means the best in point of mechanism.
The two circles are read off hy verniers, or microscopes ;
the one attached to the fixed support which carries the
principal axis, the other to an arm projecting from that
axis. Both circles also are susceptible of being clamped,
the clamps being attached to the same ultimate bearing
with which the apparatus for reading off is connected.

(148.) It is manifest that such a combination, however
its principal axis be pointed (provided that its direction
be invariable), will enable us to ascertain the situation of

any object with respect to
the observer's station, by
angles reckoned upon two
great circles in the visible
hemispliere, one of which
has for its poles the pro-
longations of the principal
axis or the vanishing points
of a system of lines parallel
to it, and the other passes
always through these poles ;
for the former great circle
is the vanishing line of all
planes parallel to the circle
AB, while the latter, in any
position of the instrument,
is the vanishing line of all
the planes parallel to the
circle GH ; and these two planes being, by the construc-
tion of the instalment, at right angles, the great circles,

98 A TREATISE ON ASTRONOMV. [cHAP. 11.

which are their vanishing lines, must be so too. Now,
if two great circles of a spliere be at right angles to each
other, the one will always pass through the other's
poles.

(149.) There are, however, but two positions in which
such an apparatus can be mounted so as to be of any
practical utility in astronomy. The first is, when the
principal axis CD is parallel to the earth's axis, and
therefore points to the poles of the heavens which are the
vanishing points of all lines in his system of parallels :
and when, of course, the plane of the circle AB is paral-
lel to the earth's equator, and therefore, has the equi-
noctial for its vanishing circle, and measures, by its arcs
read off, hour angles, or differences of right ascension.
In this case, the great circles in the heavens, correspond*
ing to the various positions, which the circle Gil can be
made to assume, by the rotation of the instrument round
its axis CD, are all hour-circles : and the arcs read off
on this circle will be declinations, or polar distances, or
their differences.

(150.) In this position the apparatus assumes the name
of an equatorial, or, as it was formerly called, a parallactic
instrument. It is one of the most convenient instruments
for all such observations as require an object to be kept
long in view, because, being once set upon the object,
it can be followed as long as we please by a single motion,
i. e. by merely turning the whole apparatus round on its
polar axis. For since, when the telescope is set on a
star, the angle between its direction and that of the polar
axis is equal to the polar distance of the star, it follows,
that when turned about its axis, without altering the posi-
tion of the telescope on the circle GH, the point to which
it is directed will always lie in the small circle of the
heavens coincident with the star's diurnal path. In many
observations this is an inestimable advantage, and one
which belongs to no other instrument. The equatorial
is also used for determining the place of an unknown by
comparison with that of a known object, in a manner to
be described in the fourth chapter. The adjustments of
the equatorial are somewhat complicated and difficult.
They are best performed by following the pole-star round

CHAP. II. J AZIMUTH AND ALTITUDE INSTRUMENT. 99

the entire diurnal circle, and by observing, at proper in-
tervals, other considerable stars whose places are well
ascertained.*

(151.) The other position in which such a compound
apparatus as we have described in art. 147 may be advan-
tageously mounted, is that in which the principal axis
occupies a vertical position, and the one circle, AB, con-
sequently corresponds to the celestial horizon, and the
other, GA, to a vertical circle of the lieavens. The an-
gles measured on the former are therefore azimuths, or
differences of azimuth, and those on the latter zenith dis-
tances, or altitudes, according as the graduation com-
mences from the upper point of its limb, or from one 90°
distant from it. It is therefore known by the name of
an azimuth and altitude instrument. The vertical posi-
tion of its principal axis is secured either by a plumb-
line suspended from the upper end, which, however it
be turned round, should continue always to intersect one
and the same fiducial mark near its lower extremity, or
by a level fixed directly across it, whose bvibble ought
not to sliift its place, on moving the instrument in azi-
muth. The north or south point on the horizontal cir-
cle is ascertained by bringing the vertical circle to coin-
cide with the plane of the meridian, by the same criterion
by which tlie azimuthal adjustment of the' transit is per-
formed (art. 137), and noting, in this position, the read'
ing off of the lower circle, or by the following process.

(152.) Let a bright star be observed at a considerable

distance to the east of the meridian, by bringing it on

the cross wires of the telescope. In this position let the

horizontal circle be read off, and the telescope securely

clamped on the vertical one. When the star has passed

the meridian, and is in the descending point of its daily

course, let it be followed by moving the whole instrument

round to the west, without, however, unclamping the

telescope, until it comes into the field of view ; and, until,

by continuing the horizontal motion, the star, and the

cross of the wires come once more to coincide. In this

position it is evident the star must have the same precise

*See Littrowontlie Adjustment of the Equatorial. — Mem. Astron. So^,
vol. ii. p. 4&.

100 A TREATISE ON ASTRONOMY. {^CHAP. 11.

altitude about the western horizon, that it had at the mo-
ment of the first observation above the eastern. At this
point let the motion he arrested, and the liorizontal circle
be again read off. The difiereace of the readings will be
the azimuthal arc described in the interval. Now, it is
evident that when the altitudes of any star are equal on
either side of the meridian, its azimuths, whether reckon-
ed both from the north or both from the south point of the
horizon, must also be equal, — consequently the north or
south point of the horizon must bisect the azimuthal arc
thus determined, and will therefore become known.

(153.) This method of determining the north and
south points of a horizontal circle (by which, Avhen
known, we may draw a meridian line) is called the
" method of equal altitudes," and is of great and constant
use in practical astronomy. If we note, at the moments
of the two observations, the time, by a clock or chrono-
meter, the instant halfway between them will be the
moment of the star's meridian passage, which may thus
be determined without a transit ; and, vice versa, the
error of a clock or chronometer may by this process be
discovei'ed. For this last purpose, it is not necessary
that our instrument should be provided witli a horizontal
circle at all. Any means by which altitudes can be mea-
sured will enable us to determine the moments when the
same star arrives at equal altitudes in the eastern and
western halves of its diurnal course ; and, these once
known, the iiistaat of meridian passage and the error of
the clock become also known.

(154.) One of the chief purposes to which the altitude
and azimuth circle is applicable is the investigation of
the amount and laws of refraction. For, by following
with it a circumpolar star which passes the zenith, and
another wliich grazes the horizon, through their whole
diurnal course, the exact apparent form of their diurnal
orbits, or the ovals into whicli their circles are distorted
by refraction, can be traced ; and their deviation from
circles, being at every moment given by the nature of
the observation in the direction in inhich the refraction
itself takes place (i. e. in altitude), is made a matter of
direct observation.

CHAP. 11. J hadley's sextant. 101

(155.) The zenith sector and the theGiloUte are pecu-
liar modifications of the altitude and azimuth instrument.
The former is adapted for the very exact observation of
stars in or near the zenith, by giving' a great length to
the vertical axis, and suppressing all the circumference of
the vertical circle, except a few degrees of its lower
part, by which a great length of radius, and a consequent
proportional enlargement of the divisions of its arc, is
obtained. The latter is especially devoted to the mea-
sure of horizontal angles between terrestrial objects, in
Avhich the telescope never requires to be elevated more
than a few degrees, and in which, therefore, the vertical
circle is either dispensed with, or executed on a smaller
scale, and with less delicacy ; while, on the other hand,
great care is bestowed on securing the exact perpendicu-
larity of the plane of the telescope's motion, by resting
its horizontal axis on two supports like the piers of a
transit-instrument, while themselves are firmly bedded on
the spokes of the horizontal circle, and turn with it.

(156.) The last instrument we shall describe is one
by whose aid the direct angular distance of any two ob-
jects may be measured, or the altitude of a single one
determined, either by measuring its distance from the
visil:)le horizon (such as the sea-ofling, allowing for its
dip), or from its own reflection on the surface of mercury.
It is the sextant, or quadrant, commonly called Hadley' s^
from its reputed inventor, though the priority of invention
belongs undoubtedly to Newton, whose claims to the
gratitude of the navigator are thus doubled, by his having
furnished at once the only theory by which his vessel
can be securely guided, and the only instrument which
has ever been found to avail, in applying that theory to
its nautical uses.*

(157.) The principle of this instrument is the optical
property of reflected rays, thus announced : — " The

* Newton communicated it to Dr. Halley, who suppressed it. The
description of the instrument was found, after the death of Halley,
among his papers, in Newton's own handwriting, by his executor, who
communicated the papers to the Royal Society, twenty-five years after
Newton's death, and eleven after the publication of Hadley's invention,
which might be, and probably was, independent of any knowledge of
Newton's, though Hutton insitiuates the contrary.

103 A TREATISE ON ASTRONOMY. [CHAP. 11.

angle between the first and last directions of a ray which
has suffered two reflections in one plane is equal to twice

the inclination of the reflecting' surfaces to each other."
Let AB be tlie liinb, or graduated arc, of a portion of a
circle 60° in extent, but divided into 120 equal parts.
On the radius CB let a silvered plane glass D be fixed,
at riglit angles to the plane of the circle, and on the
moveable radius CE let another such silvered glass, C,
be fixed. The glass D is permaneatl^' fixed parallel to
AC, and only one half of it is s'dveied, the other half
allowing objects to be seen through it. The glass C is
wholly silvered, and its plane i.s parallel to the length
of the moveable radius CE, at the extremity E, of which
a vernier is placed to read ofl' the divisions of the limb.
On the radius AC is set a telescope F, through which
any object, Q, may be seen by direcl rays Avhich pass
through the unsilvered portion of the glass D, while
another object, P, is seen through the same telescope
by rays, which, after reflection at C, have been thrown
upon the silvered part of D, and are thence directed by
a second reflection into the telescope. The two images
so formed will both be seen in the field of view at once,
and by moving the radius CE will (if the reflectors be
truly perpendicular to the plane of the circle) meet and
pass over, without obliterating each other. The motion,
however, is arrested when they meet, and at this point
the angle included between the direction CP of one
object, and FQ of the other, is twice the angle ECB in-
cluded between the fixed and moveable radii CB, CE.
Now the graduations of the limb being purposely made

CHAP. II.] PRINCIPLE OF REPETITION. 103

only half as distant as would correspond to degrees, the
arc BE, when read off, as if the graduations were whole
degrees, will, in fact, read double its real amount, and
therefore tlie numbers to read off will express not the
angle ECB, but its double, the angle subtended by the
objects.

(158.) To determine the exact distances between the
stars by direct observation is comparatively of little ser-
vice ; but in nautical astronomy the measurement of
their distances from the moon, and of their altitudes, is
of essential importance ; and as the sextant requires no
fixed support, but can be held in the hand, and used on
ship-board, the utility of the instrument becomes at once
obvious. For altitudes at sea, as no level, plumb-line,
or artificial horizon can be used, the sea-offing affords
the only resource ; and the image of the star observed,
seen by reflection, is brought to coincide with the boun-
dary of the sea seen by direct rays. Thus the altitude
above the sea-line is found ; and this corrected for the
dip of the horizon (art. 24) gives the true altitude of the
star. On land, an artificial horizon may be used (art. 139),
and the consideration of dip is rendered unnecessary.

(159.) The reflecting circle is an instrument destined
for the same uses as the sextant, but more complete, the
circle being entire, and the divisions carried all round.
It is usually furnished with three verniers, so as to admit
of three distinct readings off, by the average of which
the error of graduation and of reading is reduced. This
is altogether a very refined and elegant instrument.

(160.) We must not conclude this chapter without
mention of the " principle of repetition ;" an invention
of Borda, by which the error of graduation may be di-
minished to any degree, and, practically speaking, anni-
hilated. LetPQ be two objects which we may suppose
fixed, for purposes of mere explanation, and let KL be a
telescope moveable on O, the common axis of two cir-
cles, AML and a 6 c, of which the former, AML, is ab-
solutely fixed in the plane of the objects, and carries the
graduations, and the latter is freely moveable on the axis.
The telescope is attached permanently to the latter circle,
and moves with it. An arm OaA carries the index, or

104 A TREATISE ON ASTRONOMY. [cHAP. H.

vernier, which reads off the graduated limb of the fixed
circle. This arm is provided with two clamps, by which
it can be temporarily connected with either circle, and

JEC JO

detached at pleasure. Suppose, now, the telescope di-
rected to P. Clamp the index arm OA to the inner
circle, and unclamp it from the outer, and read off. Then
carry the telescope round to the other object Q. In so
doing, the inner circle, and the index-arm which is
clamped to it, will also be carried I'ound, over an arc AB,
on the graduated liml? of the outer, equal to the angle
POQ. Now clamp the index to the outer circle, and
unclamp the inner, and read off: the difference of readings
will of course measure the angle POQ ; but the result
will be liable to two sources of error — that of graduation
and that of observation, both which it is our object to
get rid of. To this end transfer the telescope back to P,
without unclamping the arm from the outer circle; then,
having made the bisection of P, clamp the arm to b, and
unclamp it from B, and again transfer the telescope to Q,
by which the arm will now be carried with it to C, over
a second arc, BC, equal to the angle POQ. Now again
yead off ; then will the difference between this reading
and the original one measure twice the angle POQ,
affected with both errors of observation, but only with
the same error of graduation as before. Let this pro-
cess be repeated as ofteii as we please (suppose ten
times) ; then will the final arc ABCD read off on the
eircl? be ten times the required angle, affected by the.

CHAP. III. J GEOGRAPHY. 105

joint errors of all the ten observations, but only by the
same constant error of graduation, which depends on the
initial and final readings off alone. Now the errors of
observation, when numerous, tend to balance and destroy
one another ; so that, if sufficiently multiplied, their in-
fluence will disappear from the result. There remains,
then, only the constant error of graduation, which comes
to be divided in the final result by the number of obser-
vations, and is therefore diminished in its influence to
one tenth of its possible amount, or to less if need be.
The abstract beauty and advantage of this principle seem
to be counterbalanced in practice by some unknown
cause, which, probably, must be sought for in imperfect
clamping.

CHAPTER III.

OF GEOGRAPHY.

Of the FigTirp of the Earth — Tts exact Dimensinns — Its Form that of Equi-
librium modiiicd by centrifugal Force — Variation of Gravity on its
Surface — Statical and Dynamical Measures of Gravity — The Pendu-
lum— Gravity to a Spheroid — Other Effects of Earth's Rotation — Trade
Winds — Determination of geographical Positions — Of Latitudes — Of
Longitudes — Conduct of a trigonometrical Survey — Of Maps — Pro-
jections of the Sphere — Measurement of Heights' by the Barometer.

(161.) Geography is not only the most important of
the practical l)ranehes of knowledge to Avhieh astronomy
is applied, but is also, theoretically speaking, an essen-
tial part of the latter science. The earth being the ge-
neral station from which we view the heavens, a know-
ledge of the local situation of particular stations on its
surface is of great consequence, when we come to inquire
the distances of the nearer heavenly bodies from us, as
concluded from observations of their parallax as well as
on all other occasions, where a difference of locality can
be supposed to influence astronomical results. We pro-
pose, therefore, in this chapter, to explain the principles
by which astronomical observation is applied to geo-
graphical determinations, and to give at the same time

106 A TREATISE ON ASTRONOMY. [cHAP. III.

fin outline of geography so far as it is to be considered a
part of astronomy.

(1G2.) Geography, as the word imports, is a delinea-
tion or description of the earth. In its widest sense, this
comprehends not only the delineation of the form of its
continents and seas, its rivers and mountains, but their
physical condition, climates, and products, and their
appropriation by com.munities of men. With physical
^nd political geography, however, we have no concern
here. Astronomical geography has for its objects the
exact knowledge of the form and dimensions of the earth,
the parts of its surface occupied by sea and land, and the
configuration of the surface of the latter, regarded as pro-
tuberant above the ocean, and broken into the various
forms of mountain, table land, and valley ; neither should
the form of the bed of the ocean, regarded as a continua-
tion of the surface of the land beneath the water, be left
out of consideration ; we know, it is true, very little of
it ; but this is an ignorance rather to be lamented, and,
if possible, remedied, than acquiesced in, inasmuch as there
are many very important branches of inquiry which would
be greatly advanced by a better acquaintance with it.

(163.) With regard to the figure of the earth as a
whole, we have already shown that, speaking loosely, it
may be regarded as spherical ; but the reader who has
duly appreciated the remarks in art. 23 will not be at a
loss to perceive that this result, concluded from observa-
tions not susceptible of much exactness, and embracing
very small portions of the surface at once, can only be
regarded as a first approximation, and may require to be
materially modified by entering into minutiae before neg-
lected, or by increasing the delicacy of our observations,
or by including in their extent larger areas of its surface.
For instance, if it should turn out (as it will), on minuter
inquiry, that the true figure is somewhat elliptical, or
flattened, in the manner of an orange, having the diame-
ter which coincides with the axis about gi^th part shorter
than the diameter of its equatorial circle ; this is so
trifling a deviation from the spherical form that, if a mo-
del of such proportions were turned in wood,'^and laid
before us on a table, the nicest eye or hand would not

Chap. hi. J figure of the earth. 107

detect the flattening, since the diflerence of diameters, ill
a globe of sixteen inches would amount only to jV^^ ^^
an inch. In all common parlance, and for all ordinary
purposes, then, it would still be called a globe ; while,
nevertheless, by careful measurement, the difljerence
would not fail to be noticed, and, speaking strictly, it
would be termed, not a globe, but an oblate ellipsoid, or
spheroid, which is the name appropriated by geometers
to the form above described.

(164.) The sections of such a figure by a plane are not
circles, but ellipses ; so that, on such a shaped earth, the
horizon of a spectator would nowhere (except at the
poles) be exactly circular, but somewhat elliptical. It ia
easy to demonstrate, however, that its deviation from the
circular form, arising from so very slight an " ellipticity"
as above supposed, would be quite imperceptible, not
only to our eyesight, but to the test of the dipsector ; so
that by that mode of observation we should never be led
to notice so small a deviation from perfect sphericity*
How we are led to this conclusion, as a practical result^
will appear, when we have explained the means of de-
termining with accuracy the dimensions of the whole, oi"
any part of the earth.

(165.) As Vie cannot grasp the earth, nor recede frottl
it far enough to view it at once as a whole, and compare
it with a known standard of measure in any degree com"
mensurate to its own size, but can only creep about upoil
it, and apply our diminutive measures to comparatively
small parts of its vast surface in succession, it become^
necessary to supply, by geometrical reasoning, the defect
of our physical powers, and from a delicate and careful
measurement of such small parts to conclude the form
and dimensions of the whole mass. This would present
little difficulty, if we were sure the earth were strictly a
sphere, for the proportion of the circumference of a circle
to its diameter being known (viz. that of 3* 141 5926 to
!• 0000000), we have only to ascertain the length of the
entire circumference of any great circle, such as a meri-
dian, in miles, feet, or any other standard units, to knOW
the diameter in units of the same kind. Now the cir-*
cumference of the whole circle is known as soon as w«j

108 A TREATISE ON ASTKONOMV. [niAI'. III.

know the exact lengtli of any aliquot part of it, such as
1° or 3^6 o*^li P^i"t ; and, this being not more than about
seventy miles in lengtli, is not l)eyoncl the limits of very
exact measurement, and could in fact, be measured (if
we knew its exact termination at each extremity) Avithin
a veiy few feet, or, indeed, inches, by methods presently
to be particularized.

(J 66.) Supposing, then, we were to begin measuring
with all due nicety from any station, in the exact direc-
tion of a meridian, and go measuring on, till by some in-
dication we were informed that we had accomplished an
exact degree from the point we set out from, our problem
Avould then be at once resolved. It only remains, there-
fore, to inquire by what indications we can be sure, 1st,
that we have advanced an exact degree ; and, 2dly , that Ave
have been measuring in the exact direction of a great circle.

(167.) Now, the earth has no landmarks on it to in-
dicate degrees, nor traces inscribed on its surface to guide
lis in such a course. The compass, though it affords a
tolerable guide to the mariner or the traveller, is far too
uncertain in its indications, and too little known in its
laws, to be of any use in such an operation. We must,
therefore, look outwards and refer our situation on the
surface of our globe to natural marks, external to it,
and Avhich are of equal permanence and stability with the
earth itself. Such marks are afforded by the stars. By
observations of their m'eridian altitudes, performed at any
station, and from their known polar distances, Ave con-
clude the height of the pole ; and since the altitude of the
pole is equal to the latitude of the place (art. 95), the
same observations give the latitudes of any stations Avhere
we may establish the requisite instnmients. When our
latitude, then, is found to have diminished a degree, Ave
know that, provided we have kept to the meridian, we
have described one three hundred and sixtieth part of the
earth's circumference.

(168.) The direction of the meridian may be secured
at every instant by the obserA'ations described in art, 137,
and although local difficulties may oblige us to deviate in
our measurement from this exact direction, yet if Ave
Keep a strict account of the amount of this deviation, a

CJUAP. 111.] LENGtll OF A DEGREE OF LATITUDE. 109

very simple calculation will enable us to reduce our ob->
served measure to its meridional value.

(169.) Such is the principle of that most important
geographical operation, the measurement of an arc of
the meridian. In its detail, however, a somewhat modi-
fied course must be followed. An observatory cannot be
mounted and dismounted at every step ; so that we can-
not identify and measure an exact degree neither more
nor less. But this is of no consequence, provided we
know with equal precision how much, more or less, we
have measured. In place, then, of measuring this pre-
cise aliquot part, we take the more convenient method
of measuring from one good observing station to another,
about a degree, or two or three degrees, as the case may
be, apart, and determining by astronomical observation
the precise difference of latitudes between the stations.

(170.) Again, it is of great consequence to avoid in
this operation every source of uncertainty, because an
error committed in the length of a single degree will be
multiplied 360 times in the circumference, and nearly
115 times in the diameter of the earth concluded from it.
Any error which may affect the astronomical determination
of a star's altitude will be especially influential. Now
there is still too much uncertainty and fluctuation in the
amount of refraction at moderate altitudes, not to make it
especially desirable to avoid this source of error. To
effect this, we take care to select for observation, at the
extreme stations, some star which passes through or near
the zeniths of both. The amount of refraction, within a
few degrees of the zenith, is very small, and its fluctua-
tions and uncertainty, in point of quantity, so excessively
minute as to be utterly inappreciable. Now, it is the
same thing whether we observe the pole to be raised or
depressed a degree, or the zenith distance of a star when
on the meridian to have changed by the same quantity.
If at one station we observe any star to pass through the
zenith, and at the other to pass one degree south or north
of the zenith, we are sure that the geographical latitudes,
or the altitudes of the pole at the two stations, must dif-
fer by the same amount.

(171.) Granting that the terminal points of one degree

K

110 A TREATISE ON ASTRONOMY. [cHAP. III.

can be ascertained, its length may be measured by tlie
methods which will be presently described, as we have
before remarked, to within a very few feet. Now, the
error which may be committed in fixing each of these
terminal points cannot exceed that which may be com-
mitted in the observation of the zenith distance of a star,
properly situated for the purpose in question. This error,
with proper care, can hardly exceed a single second.
Supposing we grant the possibility of ten feet of error in
the measured length of one degree, and of one second in
each of the zenith distances of one star, observed at the
northern and southern stations, and, lastly, suppose all
these errors to conspire, so as to tend all of them to give
a result greater or all less than the truth, it will appear,
by a very easy proportion, that the whole amount of
error which would be thus entailed on an estimate of the
earth's diameter, as concluded from such a measure,
would not exceed 544 yards, or about the third part of a
mile, and this would be large allowance.

(172.) This, however, supposes that the form of the
earth is that of a perfect sphere, and, in consequence, the
lengths of its degrees in all parts precisely equal. But
when we come to compare the measures of meridional
arcs made in various parts of the globe, the results ob-
tained, although they agree sufficiently to show that the
supposition of a spherical figure is not very remote from
the truth, yet exhibit discordances far greater than what
we have shown to be attributable to error of observation,
and which render it evident that the hypothesis, in strict-
ness of its wording, is untenable. The following table
exhibits the lengths of a degree of the meridian (astro-
nomically determined as above described), expressed in
British stancferd feet, as resulting from actual measure-
ment, made with all possible care and precision, by com-
missioners of various nations, men of the first eminence,
supplied by their respective governments with the best
instruments, and furnished with every facility which
could tend to insure a successful result of their import-
ant labours.*

* The first three columns of this table are extracted from among the
data given m Professor Airy's excellent paper " On the Figure of the
Earth," in the Encyclopaedia Metropolitana.

CHAP, in.] DEGREES IN DIFFERENT LATITUDES.

Ill

Country.

Latitude
of Middle
of the Arc

Arc
measured.

Length

of the

Degree

concluded.

Observers.

Sweden

C6 20 10

. 1''.3719'

3G5782

Svanberg.

Russia -

58 17 37

3 35 5

3053ti8

Struve.

England

52 35 45

3 57 13

364971

Roy, Kater.

France

4() 52 2

8 20 0

364872

Lacaille, Cassini.

France -

44 51 2

12 22 13

364535

Delambre,Mechain.

Rome

42 59 0

2 9 47

364262

Boscovich.

America, U. S. -

■Sd 12 0

1 28 45

3(j3786

Mason, Dixon.

Cape of Good Hope

33 18 30

1 13 ]7i

364713

Lacaille.

India

16 8 22

15 57 40

363044

Lambton, Everest.

India

12 32 21

1 34 5t)

36.3013

Lambton.

Peru

1 31 0

3 7 3

362808

Condaniine, &c.

It is evident from a mere inspection of the second and
fourth coUimns of this table tliat the measured length of
a degree increases with the latitude, being greatest near
the poles, and least near the equator. Let us now con-
sider what interpretation is to be put upon this conclusion,
as regards the form of the earth.

(173.) Suppose we held in our hands a model of the
earth smoothly turned in wood, it would be, as already-
observed, so nearly spherical, that neither by the eye nor
the touch, unassisted by instruments, could we detect any
deviation from that form. Suppose, too, we were debar-
red from measuring directly across from surface to surface
in different directions with any instrument, by which we
might at once ascertain whether one diameter were longer
than another ; how, then, we may ask, are we to ascer-
tain whether it is a true sphere or not ? It is clear that
we have no resource, but to endeavour to discover, by

some nicer means than simple inspection or feeling,
whether the convexity of its surface is the same in
every part ; and if not, where it is greatest, and where
least. Suppose, then, a thin plate of metal to be cut into

112 A TREATISE ON ASTROXOMY. [cHAP. in.

& concavity at its edge, so as exactly to fit the surface at
A ; let this now be removed from A, and applied succes-
sively to severarother parts of the surface, taking care to
keep its plane always on a great circle of the globe, as
here represented. If, then, we find any position, B, in
which the light can enter in the middle between the globe
and plate, or any other, G, where the latter tilts by pres-
sure, or admits the light under its ed^es, we are sure that
the curvature of the surface at B is less, and at C greater
than at A.

(174.) What we here do by the application of a metal
plate of determinate length and curvature, we do on the
earth by the measurement of a degree of variation in the
altitude of the pole. Curvature of a surface is nothing
but the continual deflection of its tangent from one fixed
direction as we advance along it. When, in the same
Pleasured distance of advance, we find the tangent
(which answers to our horizon) to have shifted its posi-
tion with respect to a fixed direction in space (such as
the axis of the heavens, or the line joining the earth's
eentre find some given star), mo7'e in one part of the
earth's meridian than in another, we conclude, of ne-
cessity, that the curvature of the surface at the former
spot is greater than at the latter ; and, vice versa, Avhen,
in order to produce the same change of horizon with
respect to the pole (suppose 1°), we require to travel
over a longer measured space at one point than at an-
other, we assign to that point a less curvature. Hence
we conclude that the curvature of a meridional section
&f ihe earth is sensibly greater at the equator than to-
wards the poles ; or, in other words, that the earth is
not spherical, but flattened at the poles, or, which comea
io the same, protubpi-ant at the equator.

(175.) Let NABDEF represent a meridional section
of the eartli, C its centre, and NA, BD, GE, arcs of a
meridian, each corresponding to one degree of difference
of latitude, or to one degree of variation in the meridian
altitude of a star, as referred to the horizon of a spectator
travelling along the meridian. Let nN, ak, bB, rfD, ^'•G,
«E, be the respective directions of ihe plumb-line at" the
stations N, A, B, D, G, E, of which we will suppose N

CHAP, in,] MERIDIONAL SECTION OF THE EARTH.

113

to be at the pole and E at the equator ; then will tlie tan-
gents to the surface at these pouits respectively bo per-
pendicular to these directions ; and, consequently, if each
pair, viz. nN and oA, 6B and (ID, gG and eE, be pro-
longed till they intersect each other (at the points x, y, z),
the angles N.rA, B^D, OzE, will each be one degree,
and, therefore, all equal ; so that the small curvilinear
arcs NA, BD, GE, may be regarded as arcs of circles
of one degree each, described about x, y, z, as centres.
These are what in geometry are called centres of curva-
ture, and the radii a:N or xA, yB or yl), zG or zE,, re-
present 7'adii of curvature, by which the curvatures at
those points are determined and measured. Now, as the
arcs of different circles, which subtend equal angles at
their respective centres, are in the direct proportion of
their radii, and as the arc NA is greater than BD, and
that again than GE, it follows that the radius Nx must
be greater than By, and By than Ez. Thus it appear*
that the mutual intersections of the plumb-lines will not,
as in the sphere, all coincide in one point C, the centre,
but will be arranged along a certain curve, xyz (which
will be rendered more evident by considering a number
of intermediate stations). To this curve geometers have
given the name of the evolute of the curve NABDGE,
from whose centres of curvature it is constructed.

k2

114 A TREATISE ON ASTRONOMY. [cHAP. III.

(176.) In the flattening of a round figure at two op-
posite points, and its protuberance at points rectangularly-
situated to the former, we recognise the distinguishing
feature of the elliptic form. Accordingly, the next and
simplest supposition that we can make respecting the
nature of the meridian, since it is proved not to be a
circle, is, that it is an ellipse, or nearly so, having NS,
the axis of the earth, for its shorter, and EF, the equa-
torial diameter, for its longer axis ; and that the form of
the earth's surface is that which would arise from making
such a curve revolve about its shorter axis NS. This
agrees well with the general course of the increase of
the degree in going from the equator to the pole. In the
ellipse, the radius of curvature at E, the extremity of the
longer axis is the least, and at that of the shorter axis,
the greatest it admits, and the form of its evolute agrees
with that here represented.* Assuming, then, that it is
an ellipse, the geometrical properties of that curve ena-
ble us to assign the proportion between the lengths of its
axes which shall correspond to any proposed rate of va-.
nation in its curvature, as well as to fix; upon their ab-
solute lengths, con-esponding to any assigned length of
the degree in a given latitude. Without troubling the
reader with the investigation (which may be found in
any work on the conic sections), it will be sufficient to
state that the lengths which agree on the whole best with
the entire series of meridional arcs which have been
satisfactorily measured, are as follow :t —

Feet. Miles.

Greater ox erjiialorial d iameter = 41 ,847,426 =7925-648

Le.sser or polar diameter" = 41,707,620= 7899170

Diftererif-e of diameters, or polar com- > _ 139306= 26478
pression S

Tlie proportion of the diameters is very nearly that of
298 : 299, and their difference ^^y of the greater, or a
very little greater than g-l^.

(177.) Thus we see that the rough diameter of 8000
miles we have hilhcrlo used is rather too great, the ex-
cess being about 100 miles, or ^\\h part. We consider
it extremely improbable that an error to the extent of

*The dotted lines are the portions of the evolute belonging to the othef
quadrants,
t See Profess. Ally's Essay before cited.

CHAP. in. J EXACT DIMENSIONS OF THE EARTH. 115

five miles can subsist in the diameters, or an uncertainty
to that of a tenth of its whole quantity in the com-
pression just stated. As convenient numbers to remem-
ber, the reader may bear in mind, that in our latitude
there are just as many thousands of feet in a degree of
the meridian as there are days in the year (365) : that,
speaking loosely, a degree is about 70 British statute
miles, and a second about 100 feet; and that the equa-
torial circumference of the earth is a little less than
25,000 miles (24,899).

(178.) The supposition of an elliptic form of the
earth's section through the axis is recommended by its
simplicity, and confirmed by comparing the numerical
results we have just set down with those of actual mea-
surement. When this comparison is executed, discord-
ances, it is true, are observed, which, although still too
great to be referred to error of measurement, are yet so
small, compared to the errors which would result from
the spherical hypothesis, as completely to justify our
regarding the earth as an ellipsoid, and referring the
observed deviations to either local or, if general, to com-
paratively small causes.

(179.) Now, it is highly satisfactory to find that the
general elliptical figure thus practically proved to exist,
is precisely what ought theoretically to result from the
rotation of the earth on its axis. For, let us suppose
the earth a spliere, at rest, of uniform materials through-
out, and externally covered with an ocean of equal depth
in every part. Under such circumstances it would ob-
viously be in a state of equilibrium ; and the water on
its surface would have no tendency to run one way or
the other. Suppose, now, a quantity of its materials
were taken from the polar regions, and piled up all
around the equator, so as to produce that difference of
the polar and equatorial diameters of 26 miles which we
know to exist. It is not less evident that a mountain
ridge or equatorial continent, only, would be thus form-
ed, from which the water would run down to the ex-
cavated part at the poles. However solid matter might
rest where it was placed, the liquid part, at least, would
not remain there, any more than if it were thrown on
the side of a hill. The consequence, therefore, would

116 A TREATISE ON ASTRONOMY. [cHAP. III.

be the formation of two great polar seas, liemmed in all
round by equatorial land. Now, this is by no means
the case in nature. The ocean occupies, indifferently,
all latitudes, with no more partiality to the polar than to
the equatorial. Since, then, as we see, the water oc-
cupies an elevation above the centre no less than 13
miles greater at the equator than at the poles, and yet
manifests no tendency to leave the former and run to-
wards the latter, it is evident that it must be retained in
that situation by some adequate poiver. No such power,
however, would exist in the case we have supposed,
which is therefore not conformable to nature. In other
words, the spherical form is not the Jigure of equili-
brium ; and therefore the earth is either not at rest, or
is so internally constituted as to attract the water to its
equatorial regions, and retain it there. For the latter
supposition there is no prima facie probability, nor any
analogy to lead us to such an idea. The former is in
accordance with all the phenomena of the apparent
diurnal motion of the heavens ; and, therefore, if it will
furnish us with the potcer in question, we can have no
hesitation in adopting it as the true one.

(180.) Now, every body knows that
when a weight is whirled round, it ac-
quires thereby a tendency to recede
from the centre of its motion ; which is
called the centrifugal force. A stone
whirled round in a sling is a common
illustration ; but a better, for our pre-
sent purpose, will be a pail of water, sus-
pended by a cord, and made to spin
round, while the cord hangs perpendi-
cularly. The surface of the water, in-
stead of remaining horizontal, will be-
come concave, as in the figure. The
centrifugal force generates a tendency in
all the water to leave the axis, and
press towards the circumference ; it is,
therefore, urged against the pail, and
forced up its sides, till the excess of
height, and consequent increase of pres-
sure downwards, just counterbalances its

CHAP, III,] ACTION OF THE SEA ON THE LAND. 117

centrifugal force, and a state of eqiii/ibrmin is attained.
The experiment is a very easy and instrnctive one, and
is admirably calculated to show how \he foi'tn of equili-
brium accommodates itself to varying circumstances.
If, for example, we allow the rotation to cease by degrees,
as it becomes slower we shall see the concavity of the water
regularly diminisli ; the elevated outward portion will de-
scend, and the depressed central rise, while all the time a
perfectly smooth surface is maintained, till the rotation is
exliausted, when the water resumes its horizontal state.

(181,) Suppose, then, a globe, of the size of tlie earth,
at rest, and covered with a uniform ocean, were to be set
in rotation about a certain axis, at first very slowly, but
by degrees more rapidly, till it turned round once in
twenty-four hours ; a centrifugal force would be thus gene-
rated, whose general tendency would be to urge the water
at every point of the surface to reeede from the axis,
A rotation might, indeed, be conceived so swift as io flirt
the whole ocean from the surface, like Avater from a mop.
But this would require a far greater velocity than what
we now speak of. In the case supposed, the iveight
of the Avater would still keep it on the earth : and the
tendency to recede from the axis could only be satisfied,
therefore, by the Avater leaving the poles, and floAving
towards the equator; there heaping itself up in a ridge,
just as the water in our pail accumulates against the side ?
and being retained in opposition to its weiglit, or natural
tendency towards the centre, by the pressure thus caused.
This, hoAvever, could not take place Avithout laying dry
the polar portions of the land in the form of immensely
protuberant continents ; and the difference of our supposed
cases, therefore, is this:— 'in the former, a gi-eat equato-
rial continent and polar seas Avould be formed 5 in the
latter, protuberant land Avould appear at the poles, and a
zone of ocean be disposed around the equator. This
would be the first or immediate effect. Let us noAV see
what Avould afterAvards happen, in the two cases, if things
were alloAved to take their natural course.

(182.) The sea is constantly beating on the land,
grinding it doAvn, and scattering its worn off particles and
fragments, in the state of mud and pebbles, over its bed,

118 A TREATISE ON ASTRONOMY. [ciIAP. III.

Geological facts afford abundant proof that the existing
continents have all of them undergone this process, even
more than once, and been entirely torn in fragments, or
reduced to powder, and submerged and reconstructed
Land, in this view of the subject, loses its attribute of
fixity. As a mass it might hold together in opposition
to forces which the water freely obeys ; but in its state
of successive or simultaneous degradation, when dissemi-
nated through the water, in the state of sand or mud, it
is subject to all the impulses of that fluid. In the lapse
of time, then, the protuberant land in both cases would
be destroyed, and spread over the bottom of the ocean,
filling up the lower parts, and tending continually to re-
model the surface of the solid nucleus, in correspondence
with the /or»i of eqidUbrium in both cases. Thus, after
a sufficient lapse of time, in the case of an earth at rest,
the equatorial continent, thus forcibly constructed, would
again be levelled and transferred to the polar excavations,
and the spherical figure be so at length restored. In
that of an earth in rotation, the polar protuberances
would gradually be cut down and disappear, being trans-
ferred to the equator (as being then the deepest sea), till
the earth would assume by degrees the form we observe
it to have — that of a flattened or oblate ellipsoid.

(183.) We are far from meaning here to trace the pro-
cess by which the earth really assumed its actual form ;
all we intend is, to show that this is the form to which,
under the condition of a rotation on its axis, it must tendf
and which it would 'attain, even if originally and (so to
speak) perversely constituted otherwise.

(184.) But, further, the dimensions of the earth and
the time of its rotation being known, it is easy thence to
calculate the exact amount of the centrifugal force,*
which, at the equator, appears to be -^-^ th part of the
force or weight by which all bodies, whether solid or
liquid, tend to fall towards the earth. By this fraction
of its weight, then, the sea at the equator is lightened,
and thereby rendered susceptible of being supported at a
higher level, or more remote from the centre than at the
poles, where no such counteracting force exists ; and
* See Cab. Cyc. Mechanics, c. viii.

CHAP. III. J LOCAL VARIATION OF GRAVITY. 119

where, in consequence, the water may be considered as
specifically heavier. Taking this principle as a guide,
and combining it with the laws of gravity (as developed
by Newton, and as hereafter to be more fully explained),
mathematicians have been enabled to investigate, a pri'
ori, what would be the figure of equilibrium of such a
body, constituted internally as we have reason to believe
the earth to be ; covered wholly or partially with a fluid;
and revolving uniformly in twenty-four hours ; and the
result of this inquiry is found to agree very satisfactorily
with what experience shows to be the case. From their
investigations it appears that the form of equilibrium is,
in fact, no other than an oblate ellipsoid, of a degree of
ellipticity very nearly identical with what is observed,
and which would be no doubt accurately so, did we know
the internal constitution and materials of the earth.

(185.) The confirmation thus incidently furnished, of
the hypothesis of the earth's rotation on its axis, cannot
fail to strike the reader. A deviation of its figure from
that of a sphere was not contemplated among the original
reasons for adopting that hypothesis, which was assumed
solely on account of the easy explanation it ofl'ers of the
apparent diurnal motion of the heavens. Yet we see
that, once admitted, it draws with it, as a necessary con-
sequence, this other remarkable phenomenon, of which
no other satisfactory account could be rendered. Indeed,
so direct is their connexion, that the ellipticity of the
earth's figure was discovered and demonstrated by New-
ton to be a consequence of its rotation, and its amount
actually calculated by him, long before any measurements
had suggested such a conclusion. As we advance with
our subject, we shall find the same simple principle
branching out into a whole train of singular and import-
ant consequences, some obvious enough, others which
at first seem entirely unconnected with it, and which,
until traced by Newton up to this their origin, had
ranked among the mest inscrutable arcana of astronomy,
as well as among its grandest phenomena.

(186.) Of its more obvious consequences, we may here
mention one which falls in naturally with our present
subject. If the earth really revolve on its axis, this rota-

120 A TREATISE ON ASTRONOMY. [cHAP. 111.

tion must generate a centrifugal force (see art. 184), the
effect of which must of course be to counteract a certain
portion of the weight of every body situated at the equa-
tor, as compared with its weight at the poles, or in any
intermediate latitudes. Now, this is fully confirmed by
experience. There is actually observed to exist a differ^
ence in the gravity, or downward tendency, of one and
the same body, when conveyed successively to stations
in different latitudes. Experiments made with the gi-eat-
est care, and in every accessible part of the globe, have
fully demonstrated the fact of a regular and progressive
increase in the weiglits of bodies corresponding to the
increase of latitude, and fixed its amount and the law of
its progression. From these it appears, tliat the extreme
amount of this variation of gravity, or the difference be-
tween the equatorial and polar weights of one and the
same mass of matter, is one part in 194 of its whole
weight, the rate of increase in travelling from the equa-
tor to the pole being as the square of the sine of the lati-'
tude.

(187.) The reader will here naturally inquire, what is
meant by speaking of the same body as having different
weights at different stations ; and, how such a fact, if
true, can be ascertained. When we weigh a body by a
balance or a steelyard we do but counteract its weight by
the equal weight of another body under the very same
circumstances ; and if both the body weighed and its
counterpoise be removed to another station, their gravity,
if changed at all, will be changed equally, so that they
will still continue to counterbalance each other. A dif-
ference in the intensity of gravity could, therefore, never
be detected by these means ; nor is it in this sense that
M'e assert that a body weighing 194 pounds at the equa-
tor will weigh 195 at the pole. If counterbalanced in a
scale or steelyard at the former station, an additional
pound placed in one or other scale at the latter Avould
inevitably sink the beam.

(188.) The meaning of the proposition may be thus ex-
plained : — Conceive a weight x suspended at the equator
by a string without weight passing over a pulley, x4, and
conducted (supposing such a thing possible) over other

CHAP. III.J STATICAL MEASURE OF GRAVITY.

1:^1

pulleys, such as B, round the earth's convexity, till the
other end hung down at the pole, and there sustained the
weight y. If, then, the weights x
and y were such as, at any one sta-
tion, equatorial or polar, would ex-
actly counterpoise each other on a
balance or when suspended side
by side over a single pulley, they
would not counterbalance each
other in this supposed situation, but"
the polar -weight^/ would preponde-
rate ; and to restore the equipoise
the weight x must be increased byy-ijth part of its quantity*
(189.) The means by which this variation of gravity
may be shown to exist, and its amount measured, are
twofold (like all estimations of mechanical power), stati-
cal and dynamical. The former consists in putting the
gravity of a weight in equilibrium, not Avith that of an-
other weight, but with a natural power of a different kind
not liable to be affected by local situation. Such a power
is the elastic force of a spring. Let ABC be a strong
support of brass standing on the foot AED cast in one
piece with it, into which is let a
smooth plate of agate, D, which can
be adjusted to perfect horizontality
by a level. At C let a spiral spring
G be attached, which carries at its
lower end a weight F, polished and
convex below. The length and
strength of the spring must be so ad-
justed that the weight F shall be sus-
tained by it just to swing clear of
contact with the agate plate in the
highest latitude at which it is intend-
ed to use the instrument. Then, if
small weights be added cautiously, it
may be made to descend till it just
gropes the agate, a contact which can _
be made with the utmost imaginable feniiililii^llil'igiiiiHiiiiliiiit
delicacy. Let these weights be noted ; the weight F de-
tached ; the spring G carefully lifted off its hook, and

122 A TREATISE ON ASTRONOMY. [CHAP. 111.

secured, for travelling, from rust, strain, or disturbance,
and the whole apparatus conveyed to a station in a lower
latitude. It will then be found, on remounting it, that,
although loaded with the same additional weights as be-
fore, the weight F will no longer have power enough
to stretch the spring to the extent required for producing
a similar contact. More weights will require to be add-
ed ; and the additional quantity necessary will, it is evi-
dent, measure the difference of gravity between the two
stations, as exerted on the whole quantity of pendent
matter, i. e. the sum of the weig"ht of F and half that
of the spiral spring itself. Granting that a spiral spring
can be constructed of such strength and dimensions
that a weight of 10,000 grains, including its own, shall
produce an elongation of 10 inches without permanently
straining it,* one additional grain Avill produce a further
extension of ^Jogth of an inch, a quantity which cannot
possibly be mistaken in such a contact as that in question.
Thus we should be provided with the means of mea-
suring the power of gravity at any station to within
Toi 00"^^ of its whole quantity.

(100.) The other, or dynamical process, by which the
force urging any given weight to the earth may be de-
termined, consists in ascertaining the velocity imparted
by it to the weight when suffered to fall freely in a given
time, as one second. This velocity cannot, indeed, be
directly measured ; but indirectly, the principles of me-
chanics furnish an easy and certain means of deducing it,
and, consequently, the intensity of gravity, by observing
the oscillations of a pendulum. It is proved in mecha-
nics (see Cab. Cyc, Mechanics, 216), that, if one and
the same pendulum be made to oscillate at different sta-
tions, or under the influence of difierent forces, and the
numbers of oscillations made in the same time in each

* Whether the process above described could ever be so far perfected
and refined as to become a substitute for the use of the pendulj- j, must
depend on the degree of permanence and uniformity of action 6 dynngs,
on the constancy or variabihty of the effect of temperature, on their elas-
tic force, on the possibility of transporting them, absolutely unaltered,
from place to place, &c. The great advantages, however, which such
an apparatus and mode of observation would nosse^'" f'. p • ' jf conve-
nience, cheapness, portability, and expedition, over th« pre>'- ■' 1-vborious,
tedious, and expensive process, render the atteippiw6jiA«"5tn making.

CHAP. III. 3 GRAVITY ON A SPHEROID. 123

case be coimtcd, the intensities of the forces will be to
each other inversely as the squares of the numbers of
oscillations made, and thus their proportion becomes
known. For instance, it is found that, under the equa-
tor, a pendulum of a certain form and length makes
86,400 vibrations in a mean solar day ; and that, Avhen
transported to London, the same pendulum makes 86,535
vibrations in the same time. Hence we conclude, that
the intensity of the force virging the pendulum down-
wards at the equator is to that at London as 86400 to
865^5, or as 1 to 1-00315; or, in other Avords, that a
mass of matter at the equator weighing 10,000 pounds "
exerts the same pressure on the ground, and the same
effort to crush a body placed below it, that 10,031 1 of the
same pounds, transported to London, would exert there.

(191.) Experiments of this kind have been made, as
above stated, with the utmost care and minutest precaution
to insure exactness in all accessible latitudes ; and their
general and final result has been, to give -j-J-^ for the frac-
tion expressing the difference of gravity at the equator
and poles. Now, it will not fail to be noticed by the
the reader, and will, probably, occur to him as an objec-
tion against the explanation here given of the fact by the
earth's rotation, that this differs materially from the frac-
tion 2-g-9 expressing tlie centrifugal force at the equator.
The difference by which the former fraction exceeds the
latter is ji-^, a small quantity in itself, but still far too
large, compared with the others in question, not to be
distinctly accounted for, and not to prove fatal to this ex-
planation if it will not render a strict account of it.

(192.) The mode in which this difference arises af-
fords a curious and instructive example of the indirect
influence which mechanical causes often exercise, and
of which astronomy furnishes innumerable instances.
The rotation of the earth gives rise to the cenljpfugal
forc^ *^ centrifugal force produces an ellipticity in the
form O' lie earth itself; and this very ellipticity of form
modifier its power of attraction on bodies placed at its
surface, and thus gives rise to the difference in question.
Here, tlien, we liave tlie same cause exercising at once a
direct ati'l an indirect influence. The amount of the former

124 A TREATISE ON ASTRONOMY. [cHAP. III.

is easily calculated, that of the latter Avith far more diffi-
culty, by an intricate and profound application of geo-
metry, whose steps we cannot pretend to trace in a work
like the present, and can only state its nature and result.

(193.) The weight of a l)ody (considered as undimi-
nished by a centrifugal force) is the effect of the earth's
attraction on it. This attraction, as Newton has demon-
strated, consists, not in a tendency of all matter to any
one particular centre, but in a disposition of every parti-
cle of matter in the universe to press towards, and if not
opposed to approach to, every other. The attraction of
the earth, then, on a body placed on its surface, is not a
simple but a complex force, resulting from the separate
attractions of all its parts. Now, it is evident, that if the
earth were a perfect sphere, the attraction exerted by it
on a body any where placed on its surface, whether at
its equator or pole, must be exactly alike, for the simple
reason of the exact symmetry of the sphere in every di-
rection. It is not less evident tliat, the earth being ellip-
tical, and this symmetry or similitude of all its parts not
existing, the same result cannot be expected. A body
placed at the equator, and a similar one at the pole of a
flattened ellipsoid, stand in a different geometrical rela-
tion to the mass as a whole. This difference, without
entering further into particulars, may be expected to
draw Avith it a difference in its forces of attraction on the
two bodies. Calculation confirms this idea. It is a
question of purely mathematical investigation, and has
been treated Avith perfect clearness and precision by New-
ton, Maclaurin, Clairaut, and many other eminent geo-
meters ; and the result of their investigations is to show
that owing to the elliptic form of the earth alone, and in-
dependent of the centrifugal force, its attraction ought to
increase the Aveight of a body in going from the equator
to th%pole by almost exactly j-g-pth part ; which, toge-
ther with 2^-f7rth due to the centrifugal force, make up the
whole quantity, y-g-jth, observed.

(194.) Another great geographical phenomenon, Avhich
owes its existence to the earth's rotation, is that of the
trade-Avinds. These mighty currents in our atmosphere,
on Avhich so important a part of navigation depends.

CHAP, in.] THE TRADE-WINDS. 125

arise from, 1st, the unequal exposure of the earth's sur-
face to the sun's rays, by which it is unequally heated
in different latitudes ; and, 2dly, from that general law
in the constitution of all fluids, in virtue of which they
occupy a larger bidk, and become specifically lighter
when hot than when cold. These causes, combined with
the earth's rotation from west to east, afford an easy and
satisfactory explanation of the magnificent phenomena in
question.

(195.) It is a matter of observed fact, of which we
shall give the explanation farther on, that the sun is con-
stantly vertical over some one or other part of the earth
between two parallels of latitude, called the tropics, re-
spectively 23 1 ° north, and as much south of the equator;
and that the whole of that zone or belt of the earth's sur-
face included between the tropics, and equally divided
by the equator, is, in consequence of the great altitude
attained by the sun in its diurnal course, maintained at a
much higher temperature than those regions to the north
and south which lie nearer the poles. Now, the heat thus
acquired by the earth's surface is communicated to the
incumbent air, which is thereby expanded, and rendered
specifically lighter than the air incumbent on the rest of
the globe. It is, therefore, in obedience to the general
laws of hydrostatics, displaced and buoyed up from the
surface, and its place occupied by colder, and therefore
heavier air, which glides in, on both sides, along the
surface, from the regions beyond the tropics ; while the
displaced air, thus raised above its due level, and unsus-
tained by any lateral pressure, flows over, as it were,
and forms an upper current in the contrary direction, or
toward the poles ; which, being cooled in its course, and
also sucked down to supply the deficiency in the extra-
tropical regions, keeps us thus a continual circulation.

(196.) Since the earth revolves about an axis passing
through the poles, the equatorial portion of its surface
has the greatest velocity of rotation, and all other parts
less in the proportion of the radii of the circles of lati-
tude to which they correspond. But as the air, when
relatively and apparently at rest on any part of the earth's
surface, is only so because in reality it participates in the

l2

126 A TREATISE ON ASTRONOMY. [CHAP. III.

motion of rotation proper to that part, it follows that
when a mass of air near the poles is transferred to the
region near the equator by any impulse urging it direct-
ly towards that circle, in every point of its progress to-
wards its new situation it must be found deficient in ro-
tatory velocity, and therefore unable to keep up with the
speed of the new surface over which it is brought.
Hence, the currents of air which set in towards the
equator from the north and south must, as they glide
along the surface, at the same time lag, or hang back,
and drag tfpon it in the direction opposite to the earth's
rotation, i. e. from east to west. Thus these currents,
which but for the rotation would be simply northerly
and southerly winds, acquire, from this cause, a relative
direction towards the west, and assume the character of
permanent north-easterly and south-easterly winds.

(197.) Were any considerable mass of air to be sud-
denly transferred from beyond the tropics to the equator,
the difference of the rotatory velocities proper to the two
situations would be so great as to produce not merely a
wind, but a tempest of the most destructive violence.
But this is not the case ; the advance of the air from tlie
north and south is gradual, and all the while the earth is
continually acting on, and by the friction of its surface
accelerating its rotatory velocity. Supposing its progress
towards the equator to cease at any point, this cause
would almost immediately communicate to it the defi-
cient motion of rotation, after which it would revolve
quietly with the earth, and be at relative rest. We have
only to call to mind the comparative thinness of the coat-
ing which the atmosphere forms around the globe (art.
34), and the immense mass of the latter, compared with
the former (which it exceeds at least 100,000,000 times),
to appreciate fully the absolute command of any exten-
sive territory of the earth over the atmosphere immedi-
ately incumlient on it, in point of motion.

(198.) It follows from this, then, that as the winds on
both sides approach the equator, their easterly tendency
must diminish.* The lengths of the diurnal circles in-

* See Captain Hall's " Fragments of Voyages and Travels," 2d series,
vol. i. p. 162, wiiere this is very distinctly, and, so far as I am aware, for
the first lime, reasoned out, — Author.

CHAP. 111.3 COMPENSATION OF THE TRADE-WINDS. 127

crease very slowly in the immediate vicinity of the equa-
tor, and for several degrees on either side of it hardly
change at all. Thus the friction of the surface has more
time to act in accelerating the velocity of the air, bring-
ing it towards a state of relative rest, and diminishing
thereby the relative set of the currents from east to west,
which, on the other hand, is feebly, and, at length, not
at all reinforced by the cause which originally produced
it. Arrived, then, at the equator, the trades must be
expected to lose their easterly character altogether. But
not only this but the northern and southern currents, here
meeting and opposing, will mutually destroy each other,
leaving only such preponderancy as may be due to a
difference of local causes acting in the two hemispheres,
which in some regions around the equator may lie one
way, in some another.

(199.) The result, then, must be the production of
two great tropical belts, in the northern of which a con-
stant north-easterly and in the southern a south-easterly,
wind must prevail, while the winds in the equatorial
belt, which separates the two former, should be compa-
ratively calm and free from any steady prevalence of
easterly character. All these consequences are agreeable
to observed fact, and the system of aerial currents above
described constitutes in reality what is understood by
the regular trade-ivinds.*

(200.) The constant friction thus produced between
the earth and atmosphere in the regions near the equator
must (it may be objected) by degrees reduce and at
length destroy the rotation of the whole mass. The
laws of dynamics, however, render such a consequence
generally impossible ; and it is easy to see, in the pre-
sent case, where and how the compensation takes place.
The heated equatorial air, while it rises and flows over
towards the poles, carries with it the rotatory velocity
due to its equatorial situation into a higher latitude,
where the earth's surface has less motion. Hence, as
it travels northward or southward, it will gain conti-
nually more and more on the surface of the earth in its
diurnal motion, and assume constantly more and more a
* See the work last cited.

128 A TREATISE ON ASTRONOMY. [cHAP. III.

westerly relative direction ; and when at length it returns
to the surface, in its circulation, which it must do more
or less in all the interval between the tropics and the
poles, it will act on it by its friction as a powerful south-
west wind in the northern hemisphere, and a north-west
in the southern, and restore to it the impulse taken up
from it at the equator. We have here the origin of the
south-west and westerly gales so prevalent in our lati--
tudes, and of the almost universal westerly winds in
the North Atlantic, which are, in fact, nothing else than
a part of the general system of the reaction of the
trades, and of the process by which the equilibrium of
the earth's motion is maintained under their action.*

(201.) In order to construct a map or model of the
earth, and obtain a knowledge of the distribution of sea
and land over its surface, the forms of the outlines of its
continents and islands, the courses of its rivers and
mountain chains, and the relative situations, with respect
to each other, of those points which chiefly interest us,
as centres of human habitation, or from other causes, it
is necessary to possess the means of determining correctly
the situation of any proposed station on its surface. For
this, two elements require to be known, the latitude and
longitude, the former assigning its distance from the
poles or the equator, the latter, the meridian on which
that distance is to be reckoned. To these, in strictness,
should be added, its height above the sea level ; but the

* As it is our object merely to illustrate the mode in which the earth's
rotation affects the atmosphere on the great scale, we omit all considera-
tion of local periodical winds, such as monsoons, &c.

It seems worth inquiry, wliether hurricanes in tropical climates may
not arise from portions of the upper currents prematurely diverted dowTi-
wards before their relative velocity has been sufficiently reduced by fric-
tion on, and gradual mixing with, the lower strata ; and so dashing upon
the earth with that tremendous velocity which gives them their destruc-
tive character, and of which hardly any rational accoimt has yet been
given. Their course, generally speaking, is in opposition to the regular
trade-wind, as it ought to be, in conformity with this idea. (Young's
Lectures, i. 704.) But it by no means follows that this must always be
the case. In general, a rapid transfer, either way, in latitude, of any
mass of air which local or temporary causes might carry above the im-
mediate reach of the friction of the earth's surface, would give a fearful
exaggeration to its velocity. Wherever such a mass should strike the
earth, a hurricane might arise ; and should two such masses encounter
in mid-air, a tornado of any degree of intensity on record might easily
result from their combination. — Author.

CHAP. III.3 GEOGRAPHICAL LATITUDES DETERMINED. 129

consideration of this had better be deferred, to avoid
complicating the subject.

(202.) Tlie latitude of a station on a sphere woukl be
merely the length of an arc of the meridian, intercepted
between tlie station and the nearest point of the equator,
reduced into degrees. (See art. 86.) But as the earth
is elliptic, this mode of conceiving latitudes becomes
inapplicable, and we are compelled to resort for our de-
finition of latitude to a generalization of that property
(art. 95), which affords the readiest means of determin-
ing it by observation, and which has the advantage of
being independent of the figure of the earth, which,
after all, is not exactly an ellipsoid, or any known geo-
metrical solid. The latitude of a station, then, is the
altitude of the elevated pole, and is, therefore, astrono-
mically determined by those methods already explained
for ascertaining that important element. In consequence,
it will be remembered that, to make a perfectly correct
map of the whole, or any part of the earth's surface,
equal difterences of latitude are not represented by ex-
actly equal intervals of surface.

(203.) To determine the latitude of a station, then, is
easy. It is otherwise with its longitude, whose exact de-
termination is a matter of more difficulty. The reason
is this : — as there are no meridians marked upon the
earth, any more than parallels of latitude, we are obliged
in this case, as in the case of the latitude, to resort to
marks external to the earth, i. e. to the heavenly bodies,
for the objects of our measurement ; but with this dif-
ference in the two cases — to observers situated at sta-
tions on the same meridian (i. e. differing in latitude)
the heavens present difierent aspects at all moments.
The portions of them which become visible in a com-
plete diurnal rotation are not the same, and stars which
are common to both describe circles diflerently inclined
to their horizons, and differently divided by them, and
attain different altitudes. On the other hand, to ob-
servers situated on the same parallel (i. e. differing only
in longitude) the heavens • present the same aspects.
Their visible portions are the same ; and the same stars
describe circles equally inclined, and similarly divided

130 A TRKATISF. ON ASTRONOMY. [cHAP. HI.

by their lionzons, and attain the same altitudes. In tho
former case there is, in the latter there is not, any thing
in the appearance of the lieavens, watched through a
whole diurnal rotation, which indicates a difference of
locality in the observer.

(204.) But no two observers, at different points of the
earth's surface, can have at the same instant the same
celestial hemisphere visible. Suppose, to fix our ideas,
an observer stationed at a given point of the equator,
and that at the moment when he noticed some bright
star to be in his zenith, and therefore on his meridian,
he should be suddenly transported, in an instant of time,
round one quarter of the globe in a ivesterhj direction, it
is evident that he will no longer have the same star ver-
tically above him : it will now appear to him to be just
rising, and he will have to wait six hours before it again
comes to his zenith, i. e. before the earth's rotation from
west to east carries him back again to the line joining
the star and the earth's centre from which he set out.

(205.) The difference of the cases, then, may be thus
stated, so as to afford a key to the astronomical solution
of the problem of the longitude. In the case of stations
differing only in latitude, the same star comes to the
meridian at the same time, but at different altittides. In
that of stations differing only in longitude, it comes to
the meridian at the same altitude, but at different times.
Supposing, then, that an observer is in possession of any
means by which he can certainly ascertain the time of a
known star's transit across his meridian, he knows his
longitude ; or if he knows the difference between its
times of transit across his meridian and across that of
any other station, he knows their difference of longitudes.
For instance, if the same star pass the meridian of a
place A at a certain moment, and that of B exactly one
hour of sidereal time, or one twenty-fourth part of the
earth's diurnal period, later, then the difierence of longi-
tudes between A and B is one hour of time or 15°, and
B is so much west of A.

(206.) In order to a perfectly clear understanding of
the principle on which the problem of finding the longi-
tude by astronomical observations is resolved, the reader

CHAP, in.] DETERMINATION OF LONGITUDES. 131

must leani to distinguish between time, in the abstract,
as common to the whole universe, and therefore reckoned
from an epoch independent of local situation, and local
time, which reckons, at each particular place, from an
epoch, or initial instant, determined by local convenience.
Of time reckoned in the former, or abstract manner, we
have an example in what we have before defined as equi-
noctial time, which dates from an epoch determined by
the sun's motion among the stars. Of the latter, or local
reckoning, we have instances in every sidereal clock in
an observatory, and in every town clock for common use.
Every astronomer regulates, or aims at regulating, his
sidereal clock, so that it shall indicate 0'' 0"' 0% when a
certain point in the heavens, called the equinox, is on the
meridian of his station. This is the epoch of his side-
real time ; Avhich is, therefore, entirely a Zoc«/ reckoning.
It gives no information to say that an event happened at
such and such an hour of sidereal time, unless we parti-
cularize the station to which the sidereal time meant
appertains. Just so it is with mean or common time.
This is also a local reckoning, having for its epoch mean
noon, or the average of all the times throughout the year,
when the sun is on the meridian of that particular
place to which it belongs ; and, therefore, in like man-
ner, when we date any event by mean time, it is neces-
sary to name the place, or particularize ivhat mean time
we intend. On the other hand, a date by equinoctial
time is absolute, and requires no such explanatory ad-
dition.

(207.) The astronomer sets and regulates his sidereal
clock by observing the meridian passages of the more
conspicuous and well known stars. Each of these holds
in the heavens a certain determinate and known place
with respect to that imaginary point called the equinox,
and by noting the times of their passage in succession by
his clock he knows when the equinox passed. At that
moment his clock ought to have marked 0" 0™ 0' ; and if
it did not, he knows and can correct its error, and by the
agreement or disagreement of the en-ors assigned by each
star he can ascertain whether his clock is correctly regu-
lated to go twenty-four hours in one diurnal period, and

132 A TREATISE ON ASTRONOMY. lCHAP. 111.

if not, can ascertain and allow for its rate. Thus, although
his clock may not, and indeed cannot, either be set cor-
rectly, or go truly, yet by applying its error and rate (as
they are technically termed), he can correct its indications,
and ascertain the exact sidereal times corresponding to
them, and proper to his locality. This indispensable
operation is called getting his local time. For simplicity
of explanation, however, we shall suppose the clock a
perfect instrument ; or, which comes to the same thing,
its error and rate applied at every moment it is consulted,
and included in its indications.

(208.) Suppose, now, two observers, at distant sta-
tions, A and B, each independently of the other, to set
and regulate his clock to the true sidereal time of his
station. It is evident that if one of these clocks could
be taken up without deranging its going, and set down by
the side of the other, they would be found, on compari-
son, to differ by the exact difference of their local epochs ;
that is, by the time occupied by the equinox, or by any
star, in passing from the meridian of A to that of B : in
other words, by their difference of longitude, expressed
in sidereal hours, minutes, and seconds.

(209.) A pendulum clock cannot be thus taken up and
transported from place to place without derangement, but
a chronometer may. Suppose, then, the observer at B
to use a chronometer instead of a clock, he may, by bodily
transfer of the instrument to the other station, procure a
direct comparison of sidereal times, and thus obtain his
longitude from A. And even if he employ a clock, yet
by comparing it first with a good chronometer, and then
transferring the latter instrument for comparison with the
other clock, the same end will be accomplished, provided
the going of the chronometer can be depended on.

(210.) Were chronometers perfect, nothing more com-
plete and convenient than this mode of ascertaining dif-
ferences of longitude could be desired. An observer,
provided with such an instrument, and with a portable
transit, or some equivalent method of determining the
local time at any given station, might, by journeying
from place to place, and observing the meridian passages
of stars at each (taking care not to alter his chronome-

CHAP. in. J LONGITUDES FOUND BY CHRONOMETERS. 133

ter, or let it run down), ascertain their difierences of lon-
gitude with any required precision. In this case, the
same time-keeper being used at every station, if, at one
of them, A, it mark true sidereal time, at any other, B,
it will be just so much sidereal time in error as the dif-
ference of longitudes of A and B is equivalent to : in
other words, the longitude of B from A will appear as the
error of the time-keeper on the local time of B. If he
travel westward, then his chronometer will appear con-
tinually to gain, although it really goes (Correctly. Sup-
pose, for instance, he set out from A, when the equinox
was on the meridian, or his chronometer at 0*", and in
twenty-four hours (sid. time) had travelled 15° westward
to B. At the moment of arrival there, his chronometer
will again point to 0'' ; but the equinox will be, not on
his new meridian, but on that of A, and he must wait
one hour more for its arrival at that of B. When it
does arrive there, then his watch will point not to O**, but
to l**, and will therefore be i** fast on the local time of
B. If he travel eastward, the reverse will happen.

(211.) Suppose an observer now to set out from any
station as above described, and constantly travelling
westward to make the tour of the globe, and return
to the point he set out from. A singular consequence
Avill happen : he will have lost a day in his reckoning
of time. He will enter the day of his arrival in his
diary as Monday, for instance, when, in fact, it is Tues-
day. The reason is obvious. Days and nights are
caused by the alternate appearance of the sun and stars,
as the rotation of the earth carries the spectator round
to view them in succession. So many turns as he
makes round the centre, so many days and nights will
he experience. But if he travel once round the globe in
the direction of its motion, he will, on his arrival, have
really made one turn more round its centre ; and if in
the opposite direction, one turn less than if he had re-
mained stationary at one point of its surface : in the
former case, then, he will have witnessed one alteration
of day and night more, in the latter one less, than if he
had trusted to the rotation of the earth aloiie to carry
him round. As the earth revolves from west to east, it

M

134 A tUEAtlBE ON ASTRONOMY. [cHAP. lit.

follows that a westward direction of his journey, by
which he counteracts its rotation, will cause him to lose
a day, and an eastward direction, by which he conspires
with it, to gain one. In the former case, all his days
will be longer ; In the latter, shorter than those of a
stationary observer. This contingency has actually hap»
pened to circumnavigators. Hence, also, it must neces'
earily happen that distant settlements, on the same meri^
dian, will differ a day in their usual reckoning of time,
according as they have been colonized by settlers arriving
in an eastward or in a westward direction, — a circum-
stance which may pi'oduce strange confusion when they
come to communicate with each other. The only mode
of correcting the ambiguity, and settling the disputes
which such a difference may give rise to, consists in
having recourse to the equinoctial date, which can never
be ambiguous.

(212.) Unfortunately for geography and navigation,
the chronometer, though greatly and indeed wonderfully
improved by the skill of modern artists, is yet far too
imperfect an instrument to be relied on implicitly. How-
ever such an instrument may preserve its uniformity of
rate for a few hours, or even days, yet in long absences
from home the chances of error and accident become so
multiplied as to destroy all security of reliance on even
the best. To a certain extent this may, indeed, be reme-
died by carrying out several, and using them as checks
on each other ; but, besides the expense and trouble, this
is only a palliation of the evil — the great and funda-
mental,— as it is the only one to which the determination
of longitudes by time-keepers is liable. It becomes ne-
cessary, therefore, to resort to other means of communi-
cating from one station to another a knowledge of its
local time, or of propagating from some principal station*
as a centre, its local time as a universal standard with
which the local time at any other, however situated, may
be at once compared, and thus the longitudes of all places
be referred to the meridian of such central point.

(213.) The simplest and most accurate method by
which this object can be accomplished, when circum-
stances admit of its adoption, is that by telegraphic signal.

CHAP. ni.J LONGITUDES DETERMINED BY SIGNALS. 135

Let A and B be two observatories, or other stations, pro-
vided with accurate means of determining their respective
local times, and let us first suppose them visible from
each other. Their clocks being regulated, and their errors
and rates ascertained and applied, let a signal be made at
A, of some sudden and definite kind, such as the flash
of gunpowder, the explosion of a rocket, the sudden ex-
tinction of a bright light, or any other which admits of
no mistake, and can be seen at great distances. The
moment of the signal being made must be noted by each
observer at his respective clock or watch, as if it were
the transit of a star, or any astronomical phenomenon,
and the error and rate of the clock at each station being
applied, the local time of the signal at each is determined.
Consequently, when the observers communicate their
observations of the signal to each other, since (owing to
the almost instantaneous transmission of light) it must
have been seen at the same absolute instant by both, the
difterence of their local times, and therefore of their
longitudes, becomes known. For example ; at A the
signal is observed to happen at 5** 0"" 0' sid. time at A,
as obtained by applying the error and rate to the time
shown by the clock at A, when the signal was seen there.
At B the same signal was seen at 5^ 4" 0% sid. time at B,
similarly deduced from the time noted by the clock atB,
by applying its error and rate, Consequently, the differ-
ence of their local epochs is 4"" 0% which is also their differ-
ence of longitudes in time, or 1° 0' 0" in hour angle.

(214.) The accuracy of the final determination may
be increased by making and observing several signals at
stated intervals, each of which afl'ords a comparison of
times, and the mean of all which is, of course, more to
he depended on than the result of any single comparison.
By this means, the error introduced by the comparison
of clocks may be regarded as altogether destroyed.

(215.) The distances at which signals can be rendered
visible must of course depend on the nature of the in-
terposed country. Over sea the explosion of rockets
may easily be seen at fifty or sixty miles ; and in moun-
tainous countries the flash of gunpowder in an open
spqon may be seeq, if a proper station be chosen for its

136 A TREATISE ON ASTRONOMY. [cHAP. III.

exhibition, at much greater distances. The interval be-
tween the stations of observation may also be increased
by causing the signals to be made not at one of them,
but at an intermediate point ; for, provided they are seen
by both parties, it is a matter of indifference where they
are exhibited. Still the interval which could be thus
embraced would be very limited, and tlie method in con-
sequence of little use, but for the following ingenious
contrivance, by which it can be extended to any distance,
and carried over any tract of country however difficult.
(216.) This contrivance consists in establishing, be-
tweeen the extreme stations, whose difference of longi-
tude is to be ascertained, and at which the local times
are observed, a chain of intermediate stations, alternately
destined for signals and for observers. Thus, let A and
Z be the extreme stations. At B let a signal station be
established, at which rockets, &c. are fired at stated in-
tervals. At C let an observer be placed, provided with
a chronometer ; at D, another signal station ; at E, an-
other observer and chronometer ; and so on till the whole

\* :'* :*

A B c D :e ji" z

line is occupied by stations so arranged, that the signals
at B can be seen from A and C ; those at D, from C and
E ; and so on. Matters being thus arranged, and the
errors and rates of the clocks at A and Z ascertained by
astronomical observation, let a signal be made at B, and
observed at A and C, and the times noted. Thus the
difference between A's clock and C's chronometer be-
comes known. After a short interval (five minutes for
instance) let a signal be made at D, and observed by C
and E. Then will the difference between their respec-
tive chronometers be determined ; and the difference
between the former and the clock at A being already as-
certained, the difference between the clock A and chro-
nometer E is therefore known. This, however, supposes

CHAP. III.3 NATURAL SIGNALS. 137

that the intermediate chronometer C has kept true side-
real time, or at least a known rate, in the interval between
the signals. Now this interval is purposely made so
very short, that no instrument of any pretension to cha-
racter can possibly produce an appreciable amount of
error in its lapse. Thus the time propagated from A to
C may be considered as handed over, without gain or
loss (save from error of observation), to E. Similarly,
by the signal made at F, and observed at E and Z, the
time so transmitted to E is forwarded on to Z ; and thus
at length the clocks at A and Z are compared. The
process may be repeated as often as is necessary to
destroy error by a mean of results ; and when the line
of stations is numerous, by keeping up a succession
of signals, so as to allow each observer to note al-
ternately those on either side, which is easily pre-
arranged, many comparisons may be kept running along
the line at once, by which time is saved, and other ad-
vantages obtained.* In important cases the process is
usually repeated on several nights in succession.

(217.) In place of artificial signals, natural ones, when
they occur sufficiently definite for observation, may be
equally employed. In a clear night the number of those
singular meteors, called shooting stars, which may be
observed, is usually very great ; and as they are sudden
in their appearance and disappearance, and from the
great height at which they have been ascertained to take
place are visible over extensive regions of the earth's
surface, there is no doubt that they may be resorted to
with advantage, by previous concert and agreement be-
tween distant observers to watch and note them.t

(218.) Another species of natural signal, of still greater
extent and universality (being visible at once over a Avhole
terrestrial hemisphere), is afibrded by the eclipses of
Jupitei-'s satellites, of which we shall speak more at large
when we come to treat of those bodies. Every such
eclipse is an event which possesses one great advantage

* For a complete account of this method, and the mode of deducing
the most advantageous result from a combinaticai of all the observations,
see a paper on the difference of longitudes of Greenwich and Paris, Phik
Trans. 1826 ; by the author of this volume.

t This idea was first suggested by the late Dr. Maskelyne,

m2

138 A TREATISE ON ASTRONOMY. [cHAP. III.

in its applicability to the purpose in question, viz. that
the time of its happening, at any fixed station, such as
Greenwich, can be predicted from a long course of pre-
vious recorded observation and calculation thereon found-
ed, and that this prediction is sufficiently precise and
certain, to stand in the place of a corresponding obser-
vation. So that an observer at any other station wher-
ever, who shall have observed one or more of these
eclipses, and ascertained his local time, instead of waiting
for a communication with Greenwich, to inform him at
what moment the eclipse took place there, may use the
predicted Greenwich time instead, and thence, at once,
and on the spot, determine his longitude. This mode
of ascertaining longitudes is, however, as will hereafter
appear, not susceptible of great exactness, and should
only be resorted to when others cannot be had. The
nature of the observation also is such that it cannot be
made at sea ; so that, however useful to the geographer,
it is of no advantage to navigation.

(219.) But such phenomena as these are of only occa-
sional occurrence ; and in their intervals, and when cut off
from all communication with any fixed station, it is indis-
pensable to possess some means of determining longi-
tudes, on which not only the geographer may rely for a
knowledge of the exact position of important stations on
land in remote regrions, but on which the navigator can
securely stake, at every instant of his adventurous course,
the lives of himself and comrades, the interests of his
country, and the fortunes of his employers. Such a me-
thod is afforded by Lunar Observations. Though we
have not yet introduced the reader to the phenomena of
the moon's motion, this will not prevent us from giving
here the exposition of the principle of the lunar method ;
on the contrary, it will be highly advantageous to do so,
since by this course we shall have to deal with the
naked principle, apart from all the peculiar sources of
difficulty with which the lunar theory is encumbered,
but which are, in fact, completely extraneous to the
principle of its application to the problem of the longi-
tudes, which is quite elementary.

(220.) If there were in the heavens a clock furnished

CHAP. 111.3 LUNAR METHOD OF LONGITUDES. 139

with a dial-plate and hands, which always marked
Greenwich time, the longitude of any station would be
at once determined, so soon as the local time was
known, by comparing it with this clock. Now, the
offices of the dial-plate and hands of a clock are these : —
the former carries a set of marks upon it, whose position
is known ; the latter, by passing over and among these
marks, informs us, by the place it holds with respect to
them, what it is o'clock, or what time has elapsed since
a certain moment wiien it stood at one particular spot.

(221.) In a clock the marks on the dial-plate are uni-
formly distributed all around the circumference of a cir-
cle, whose centre is that on wliich the hands revolve with
a uniform motion. But it is clear that we should, with
equal certainty, though with much more trouble, tell
what o'clock it were, if the marks on the dial-plate
were i<nequally distributed, — if the hands were eccentric,
and their motion not uniform, — provided we knew, 1st,
the exact intervals round the circle at which the hour
and minute marks were placed ; which would be the case
if we had them all registered in a table, from the results
of previous careful measurement : — 2dly, if we knew
the exact amount and direction of eccentricity of the
centre of motion of the hands ; — and, 3dly, if we were
fully acquainted with all the mechanism which put the
hands in motion, so as to be able to say at every instant
what were their velocity of movement, and so as to be
able to calculate, without fear of error, how much time
should correspond to so much angular movement.

(222.) The visible surface of the starry heavens is the
dial-plate of our clock, the stars are the fixed marks dis-
tributed around its circuit, tlie moon is the moveable
hand, which, with a motion that, superficially consider-
ed, seems uniform, but which, when carefully examined,
is found to be far otherwise, and regulated by mechanical
laws of astonishing complexity and intricacy in result,
though beautifully simple in principle and design, per-
forms a monthly circuit among them, passing visibly
over and hiding, or, as it is called, occulting, some, and
gliding beside and between others ; and whose position
amono" them can, at any moment when it is visible, be

140 A TREATISE ON ASTRONOMY. [^CHAP. HI.

exactly measured by the help of a sextant, just as we
imight measure the place of oui* clock-hand among the
marks on its dial-plate Avith a pair of compasses, and
thence, from the known and calculated laws of its mor-
tion, deduce the time. That the moon does so move
among the stars^ while the latter hold constantly, with
yespect to each other, the same relative position, the no-
tice of a few nights, or even hours, will satisfy the com-
mencing student, and this is all that at present we require,

(223,) There is only one circumstance wanting ta
make our analogy complete. Suppose the hands of our
clock, instead of moving quite close to the dial-plate,
were considerably elevated above, or distant in front of
it. Unless, then, in viewing it, we kept our eye just in
ihe line of their centre, we should not see them exactly
thrown or projected upon their proper places on the dial»
And if we were either unaware of this cause of optical
change of place, this parcdlax — or negligent in not
taking it into account — we might make great mistakes
in reading the time, by referring the hand to the wrong
mark, or incorrectly appreciating its distance from the
Tight. On the other hand, if we took care to note, in
every case, when we had occasion to observe the time,
the exact position of the eye, there would be no difficulty
in ascertaining and allowing for the precise influence
of this cause of apparent displacement. Now, this is;
just what obtains with the apparent motion of the moon
among the stars. The former (as will appear) is com-
paratively near to the earth — the latter immensely dis-.
tant ; and in consequence of our not occupying the cen-
tre of the earth, but being carried about on its surface, and
constantly changing place, there arises a parallax, which
displaces the moon apparently among the stars, and must
be allowed for before we can tell the true place she
would occupy if seen from the contre.

(3.24.) Such a clock as we have described might, no
doubt, be considered a very bad one ;- but if it were our
anil/ one, and if incalculable interests were at stake on
a perfect knowledge of time, we should justly regard it
as most precious, and think no pains ill bestowed in stu-
dyhig the laws of its movements, or in facilitating th©

CHAP. III. J LUNAR METHOD OF LONGITUDES. 141

means of reading it correctly. Such, in the parallel we
are drawing, is the lunar theory, whose object is to
reduce to regularity the indications of this strangely
irregular-going clock, to enable us to predict, long before-
hand, and Avith absolutely certainty, whereabouts amo;ig
the stars, at every hour, minute, and second, in every
day of every year, in Greenwich local time, the moon
ivoidd be seen from the earth's centre, and will be seen
from every accessible point of its surface ; and such is
the lunar method of longitudes. The moon's apparent
angular distances from all those principal and conspicu-
ous stars which lie in its course, as seen from the earth's
centre, are computed and tabulated with the utmost care
and precision in almanacs published under national
control. No sooner does an observer, in any part of
the globe, at sea or on land, measure its actual distance
from any one of those standard stars (whose places in
the heavens have been ascertained for the purpose with
the most anxious solicitude), than he has, in fact, per-
formed that comparison of his local time with the local
times of every observatory in the world, which enables
him to ascertain his difference of longitude from one or
all of them.

(225.) The latitudes and longitudes of any number
of points on the earth's surface may be ascertained by
the methods above described; and by thus laying down
a sufficient number of principal points, and filling in the
intermediate spaces by local surveys, might maps of
counties be constructed, the outlines of continents and
islands ascertained, the courses of rivers and mountain
chains traced, and cities and towns referred to their pro-
per localities. In practice, however, it is found simpler
and easier to divide each particular nation into a series
of great triangles, the angles of which are stations con-
spicuously visible from each other. Of these triangles,
the angles only are measured by means of the theo-
dolite, with the exception of one side only of one trian-
gle, which is called a base, and which is measured with
every refinement which ingenuity can devise or expense
command. This base is of moderate extent, rarely sur-
passing six or seven miles, and purposely selected in a

143

A TREATISE ON ASTRONOMY. [cHAP. HI.

perfectly horizontal plane, otherwise conveniently adapt-
ed for purposes of measurement. Its length between its
two extreme points (which are dots on plates of gold op
platina let into massive blocks of stone, and which are,
or at least ought to be, in all cases preserved with almost
religious car«, as monumental records of the highest im-
portance), is then measured, with every precaution to
insure precision,* and its position with respect to the
meridian, as well as the geographical positions of its ex*
tremities, carefidly ascertained.

(226.) The annexed figure represents such a chain of
triangles, AB is the base, O, C, stations visible from

both its extremities (one of which, O, we will suppose

to be a national ol)servatory, with which it is a principal

pbject that the base should be as closely and immedi-

9.tely connected as possible) ; and D, E, F, G, H, K,

Other stations, remarkable points in the county, by whose

connexion its whole surface may be covered, as it were,

■\vith a netvy^ork of triangles. Now, it is evident that tlie

angles of the triangle A, B, C being obseived, and one

of its sides, AB, measured, the other two sides, AC, BC,

^piay be calculated by the rules of trigonometry ; and thusi

each of the sides AC and BC becomes in its turn a base

capable of being employed as known sides of cHher tri^

angles. For instance, the angles of the triangles ACQ

^nd BCF being known by observation, and theii sides

AC andBC, we can thence calculate th|.lengths AG, CG,

pnd BF, CF. Again, CG and CF being known, and

the included angle GCF, GF may be calculated, and so

* The greatest possible error in the Irish base of between seven and
f^ght miles, ne^- Lonilpnderry, is sin)jTosed not to exceed two i\icUea.

doRilEctioN FOR The earth's sphericity. 143

mi. Thus may all the stations be accurately determined
and laid down, and as this process may be carried on to
any extent, a map of the whole county may be thus con*
structedj and filled in to atiy degree of detail we please.
(227.) Now, on this process there are two important
remarks to be made. The first is, that it is necessary
to be careful in the selection of stations, so as to form
triangles free from any very great inequality in their an-
gles. For instance, the triangle KBF would be a very
improper one to determine the situation of F from obser-
vations at B and K, because the angle F being very acute,
a small error in the angle K would produce a great one
in the place of F upon the line BF. Such ill-conditioned
triangles, therefore, must be avoided. But if this be at-
tended to, the accuracy of the determination of the calcu-
lated sides will not be much short of that which would
be obtained by actual measurement (were it practicable) ;
and, therefore, as we recede from the base on all sides
as a centre, it will speedily become practicable to use as
bases the sides of much larger triangles, such as GF,
GH, HK, &c. ; by which means the next step of the
operation will come to be carried on on a much larger
scale, and embrace far greater intervals, than it would
have been safe to do (for the above reason) in the imme-
diate neighbourhood of the base. Thus it becomes easy
to divide the whole face of a country into great trian^
gles of from 30 to 100 miles in their sides (according to
the nature of the ground), which, being once well deter-
mined, maybe afterwards, by a second series of subordi-
nate operations, broken up into smaller ones, and these
again into others of a still minuter order, till the final fill-
ing in is brought within the limits of personal survey and
draftsmanship, and till a map is constructed, with any
required degree of detail.

(228.) The next remark we have to make is, that all
the triangles in question are not, rigorously speaking*
plane, but spherical — existing on the surface of a sphere,
or rather, to speak correctly, of an ellipsoid. In very
small triangles, of six or seven miles in the side, thi^
may be neglected, as the difference is imperceptible 5 but
in the larger ones it must be taken into eonsidei'atiotif

144 A TREATISE ON ASTRONOMY. [cHAP. III<

It is evident that, as every object used for pointing the
telescope of a theodolite has some certain elevation, not
only above the soil, but above the level of the sea, and as,
moreover, these elevations differ in every instance, a re-
duction to the horizon of all the measured angles would
appear to be required. But, in fact, by the construction
of the theodolite (art. 155), which is nothing more than
an altitude and azimuth instrument, this reduction is made
in the very act of reading off' the horizontal angles. Let

E be the centre of the earth ;
■Lk, B, C, the places on its sphe-
rical surface, to which three
stations, A, P, Q, in a country
are referred by radii E, A,
EBP, ECQ. If a theodolite
be stationed at A, the axis of its
horizontal circle will point to E
when truly adjusted, and its
plane will be a tangent to the
sphere at A, intersecting the ra-
dii EBP, ECQ, at M and N,
above the spherical surface.
The telescope of the theodolite,
it is true, is pointed in succes-
sion to P, and Q ; but the readings off of its azimuth
circle give — not the angle PAQ between the directions
of the telescope, or between the objects P, Q, as seen
from A ; hut the azimuthcd angle MAN, which is the
measure of the angle A of tbe spherical triangle BAG.
Hence arises this remarkable circumstance, — that the sum
of the three observed angles of any of the great triangles
in geodesical operations is always found to be rather more
than 180° : were the earth's surface a plane, it ought to
be exactly 180° ; and this excess, Avhich is called the
spherical excess, is so far from being a proof of incorrect-
ness in the work, that it is essential to its accuracy, and
ofTers at the same time another palpable proof of the
earth's sphericity.

(229.) The true way, then, of conceiving the subject
of a trigonometrical survey, when the spherical form of
the earth is taken into consideration, is to regard the net-

CHAf. III.3 PROJECTIONS OF fHE BPHEIlEk 145

work of triangles with Avhich the country is coveredj as
the bases of an assemblage of pyramids converging to the
Centre of the earth. The theolodite gives us the true
measures of the angles included by the planes of these
pyrainids ; and the surface of an imaginary sphere on
the level of the sea intersects them in an assemblage of
spherical triangles, above whose angles, in the radii pro-
longed, the real stations of observation are raised, by the
superficial inequalities of mountain and valley. The ope-
rose calculations of spherical trigonometry which this
consideration would seem to render necessary for the re-
ductions of a survey, are dispensed with in practice by a
very simple and easy rule. Called the rule for the spheri-
cal excess, which is to be found in most works on trigo-
nometry.* If we would take into account the ellipticity
of the earth, it may also be done by appropriate processes
of calculation, which, howevef) are too abstruse to dwell
upon in a work like the present.

(230.) Whatever process of calculation we adopt, the
Result will be a reduction, to the level of the sea, of all the
triangles, and the consequent determination of the geo-
graphical latitude and longitude of every station observed*
Thus we are at length enabled to construct maps of
countries ; to lay down the outlines of continents and
islands; the courses of rivers ; the direction of mountain
tidges, and the places of their principal summits ; and
all those details which, as they belong to physical and
statistical, rather than to asti'onomical geography, wd
need not here dilate on. A (ew words, however, will be
necessary respecting maps, which are used as well ill
astronomy as in geography.

(231.) A map is nothing more than a fepresentation,-
tlpon a plane, of some portion of the surface of a spherCj
on which are traced the particulars intended to be ex-=
pressed, whether they be continuous outlines or points*
Now, as a spherical surface! can by no contrivance
be extended or projected into a plane, without undud

* Lardner's Trigonometry, prop. 94. Woodhouse's ditto, p. I48. 1st
edition.

t We here neglect the ellipticity of the earth, which, for such a pur-
pose as map-making, is too trifling to have any material influence.

N

146 A TREATISE ON ASTRONOMY. [cHAP. III.

enlargement or contraction of some parts in proportion
to others ; and as the system adopted in so extending or
projecting it will decide xvhat part shall be enlarged or
relatively contracted, and in what proportions ; it follows,
that when large portions of the sphere are to be mapped
down, a great difference in their representations may
subsist, according to the system of projection adopted.

(232.) The projections chiefly used in maps are the
orthographic, stereographic, and Mercator''s. In the
orthographic projection, every point of the hemisphere
is referred to its diametral plane or base, by a perpendicular
let fall on it, so that the representation of the hemisphere

thus mapped on its base, is such
as it would actually appear to
an eye placed at an infinite dis-
tance from it. It is obvious,
from the annexed figure, that
in this projection only the
central portions are represented
of their true forms, while all the exterior is more and
more distorted and crowded together as we approach the
edges of the map. Owing to this cause, the orthogra-
phic projection, though very good for small portions of
the globe, is of little service for large ones.

(233.) The stereographic projection is in great mea-
sure free from this defect. To understand this projection,

CHAP. III.] MERCATOr's PROJECTION. 147

we must conceive an eye to be placed at E, one extremity
of a diameter, ECB, of the sphere, and to view the
concave surface of the sphere, every point of which, as
P, is referred to the diametral plane ADF, perpendicular
to EB by the visual line PME. The stereographic pro-
jection of a sphere, then, is a true perspective represen-
tation of its concavity on a diametral plane ; and, as
such, it possesses some singularly elegant geometrical
properties, of which we shall state one or two of the
principal.

(234.) And first, then, all circles on the sphere are re-
presented by circles in the projection. Thus the circle
Xis projected into x. Only great circles passing through
the vertex B are projected into straiglit lines traversing
the centre C : thus, BPA is projected into CA.

2dly. Every very small triangle, GHK, on the sphere,
is represented by a similar triangle, ghk, in the projec-
tion. This is a very valuable property, as it insures a
general similarity of appearance in the map to the reality
in all its parts, and enables us to project at least a hemi-
sphere in a single map, without any violent distortion
of the configurations on the surface from their real forms.
As in the orthographic projection, the borders of the
hemisphere are imduly crowded together ; in the stereo-
graphic, their projected dimensions are, on the contrary,
somewliat enlarged in receding from the centre.

(235.) Both these projections may be considered na-
tural ones, inasmuch as they are really perspective re-

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L

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presentations of the surface on a pfene. Mercator's is
entirely an artificial one, representing the sphere as it

148 A TREATISE ON ASTRONOMY. [CHAP. HI,

cannot be seen from any one point, but as it might be
geen by an eye carried successively over every part of it,
In it, the degrees of longitude, and those of latitude^
bear always to each other their due proportion ; the
equator is conceived to be extended out into a straight
Jine, and the meridians are straight lines at right angles
to it, as in the figure. Altogether, the general character
of maps on this projection is not very dissimilar to
what would be produced by i-eferring eveiy point in the
globe to a circumscribing cylinder, by lines drawn frorn
the centre, and then unrolling the cylinder into a plane,
Jjike the stereographic projection, it gives a true repre--
gentation, as io form, of every particular small part, but
yaries greatly in point of scale in its different regions j
the polar portions in particular being extravagantly en-;
Jarged ; and the whole map, even of a single hemisphere,
ftot being con^prizable within any finite limits.

(236.) We shall not, of course, enter here into any
geographical details ; but one result of maritipie discovery
pn the great scale is, so to speak, massive enough to call
for mention as an astronomical feature. When the con-i
tinents and seas are laid down on a globe (and since the
discovery of Australia we are sui-e that no very extensive
tracts of land remain unknown, except perhaps at the
south pole), we find that it is possiljle so to divide the
globe into two hemispheres, that one shall contain nearly
<jtll the land ; the other being- almost entirely sea. It i^
|i fact, not a little interesting to Englishmen, and, com-,
tined with our insular station in that great highway of
^^ations, the Atlantic, not a little explanatory of our com-
]fnercial eminence, that London occupies nearly the centre
pf the terrestrial hemisphere. Astronomically speaking,
the fact of this divisibility of the globe into an oceanic
^nd a terrestrial hemisphere is important, as demonstra^
tive of a want of absolute equality in the density of the
^olid material of the two hemispheres. Considering the
■^hole mass of land and water as in a state of equili-
brium, it is evident that the half which protrudes must
pf necessity be buoyant : not, of course, that we mean
to assert it to be lighter than water, but, as compared
witU the whole globe, in a less degree heavier thaQ

CHAP. III. J DETERMINATION OF HEIGHTS.

149

that fluid. We leave to g-cologists to draw from these
premises their own conckisions (and Ave think them ob-
vious enough) as to the internal constitution of the globe,
and the immediate nature of the forces which sustain its
continents at their actual elevation ; but in any future
investigations which may have for their object to explain
the local deviations of the intensity of gravity, from
what the hypotliesis of an exact elliptic figure would
require, this, as a general fact, oughtnot to be lostsight of.

(237.) Our knowledge of the surface of our globe is
incomplete, unless it include the heights above the sea
level of every part of the land, and the depression of the
bed of the ocean below the surface over all its extent.
The latter object is attainable (with whatever difficulty
and however slowly) by direct sounding ; tlie former by
two distinct methods : the one consisting in trigonome-
trical measurement of tlie diff'ercnces of level of all the
stations of a survey ; the other, by the use of the baro-
meter, the principle of which is, in fact, identical with
that of the sounding line. In both cases we measure the
distance of the point whose level we would know from
the surface of an equilibrated ocean : only in the one
case it is an ocean of water ; in the other, of air. In
the one case our sounding line is real and tan^il^le ; in
the other, an imaginary one, measured by the length of
the column of quicksilver the superincumbent air is ca-
pable of counterbalancing.

(238.) Suppose that instead of air, the earth and
ocean were covered with oil, and that human life could
subsist under such circumstances. Let ABODE be a

continent, of which the portion ABO projects above the
water, but is covered by the oil, which also floats at an

n2

150 A TREATISE ON ASTRONOMY. [cHAP. jn.

uniform depth on the whole ocean. Then if we woukl
know the depth of any point D below the sea level, we
let down a plummet from F. But if we would know the
height of B above the same level, we have only to send
up a float from B to the surface of the oil ; and having
done the same at C, a point at the sea level, the difference
of the two float lines gives the height in question.

(239.) Now, though the atmosphere diff'ers from oil
jn not having a positive surface equally definite, and in
not being capable of carrying up any float adequate to
such a use, yet it possesses all the properties of a fluid
really essential to the purpose in view, and this in par-
ticular ; that, over the whole surface of the globe, its
strata of equal density are parallel to the surface of equi-.
Librium, or to what ivoidd be the surface of the sea, if
frolo)iged under the continents, and therefore each or
sny of them has all the characters of a definite surface to
measure from, pix)vided it can be ascertained and identic
fied. Now the height at which, at any station B, the
mercury in a barometer is supported, informs us at
mcc how much of the sitmosphere is incumbent on B,
or, in other words, in U'hat stratum of the general at-
mosphere (indicated by its density) B is situated ;•
whence we aj-e enabled finally to conclude, by mechani-
cal reasoning,* at what height above the sea level that
degree of density is to be found over the whole surface
of the globe. Such is the principle of the application of
the barometer to the measurement of heights. For de-»
iails, the reader is refeiTed to other works. f

(240.) Possessed of a knowledge of the heights of
stations above the sea, we may connect all stations at the
, same altitude by level lines, the lowest of which will be
the outline of the sea-coast ; and the rest will mark out,
the successive coast-lines which would take place Avere
the sea to rise by regular and equal ascensions of level
over the whole world, till the highest mountains were
submerged. The bottoms of valleys and the ridge-linea

* See Cab. Cycl. Pneumatics, art. 143.

t Biot, Astronomie Physique, vol. 3. For tables, see the work of Biot
cited. Also those of Oltmauri, annually published by the French board
of longitudes in their Annuaire: and Mr. Baily's Collection of Astrono^
»nic£«l Tables and Forinute.

CHAP. IV.] TJRANOGRAPHY. 151

of hills are determined by their property of intersecting
all these level lines at right angles, and being, subject to
that condition, the shortest and longest courses respec-
tively which can be pursued from the summit to the sea.
The former constitute the water-courses of a country ;
the latter divide it into drainage basins : and thus origi-
nate natural districts of the most ineffaceable character,
on which the distribution, limits, and peculiarities of hu-
man communities are in great measure dependent.

CHAPTER IV.

OF URANOGRAPHY.

Construction of celestial Maps and Globes by Observations of right As-
cension and Deolination — Celestial Objects distinguished into fixed
and erratic — Of the Constellations — Natural Regions in the Heavens
—The Milky Way — The Zodiac— Of the Echptic— Celestial Latitudes
and Longitudes — Precession of the Equinoxes — Nutation — Aberration
— Uranographical Prablema.

(241.) The determination of the relative situations of
objects in the heavens, and the construction of maps and
globes which shall truly represent their mutual configu-
rations, as well as of catalogues which shall preserve a
more precise numerical record of the position of -each, is
a task at once simpler and less laborious than that by
which the surface of the earth is mapped and measured.
Every star in the great constellation which appears to
revolve above us, constitutes, so to speak, a celestial sta-
tion ; and among these stations we may, as upon the
earth, triangulate, by measuring with proper instruments
their angular distances from each other, which, cleared
of the effect of refraction, are then in a state for laying
down on charts, as we would the towns and villages of a
country ; and this without moving from our place, at least
for all the stars which rise above our horizon.

(242.) Great exactness might, no doubt, be attained
by this means, and excellent celestial charts constructed ;
but there is a far simpler and easier, and, at the same
time, infinitely more accurate course laid open to us, if

152 A TREATISE ON ASTRONOMY. [cHAP. IV.

we take advantage of the earth's rotation on its axis, and
by observing each celestial object as it passes our meri-
dian, refer it separately and independently to the celes-
tial equator, and thus ascertain its place on the surface
of an imaginary sphere, which may be conceived to re-
volve with it, and on which it may be considered as pro-
jected.

(243.) The right ascension and declination of a point
in the heavens correspond to the longitude and latitude
of a station on the earth ; and the place of a star on a
celestial sphere is determined, when the former elements
are known, just as that of a town on a map, by knowing
the latter. The great advantages which the method of
meridian observation possesses over that of triangula-
tion from star to star, are, then, 1st, that in it every star
is observed in that point of its diurnal course, when it is
best seen and least displaced by refraction. 2dly, that
the instruments required (the transit and mural circle)
are the simplest and least liable to error or derangement
of any used by astronomers. 3dly, that all the observa-
tions can be made systematically, in regular succession,
and with equal advantages ; there being here no ques-
tion about advantageous or disadvantageous triangles,
&c. And, lastly, that, by adopting this course, the very
quantities which we should otherwise have to calculate
by long and tedious operations of spherical trigonometry,
and which are essential to the formation of a catalogue,
are made the objects of immediate measurement. It is
almost needless to state, then, that this is the course
adopted by astronomers.

(244.) To determine the right ascension of a celestial
object, all that is necessary is to observe the moment of
its meridian passage with a transit instrument, by a clock
regulated to exact sidereal time, or reduced to such by ap-
plying its known error and rate. The rate may be ob-
tained by repeated observations of the same star at its
successive meridian passages. The error, however, re-
quires a knowledge of the equinox, or initial point from
which all right ascensions in the heavens reckon, as lon-
gitudes do on the earth from a first meridian.

(245.") The nature of this point will be explained pre

CHAP. IV.] RIGHT ASCENSIONS AND DECLINATIONS. 153

sently ; but for the purposes of uranography, in so far as
they concern only the actual configurations of the stars
inter se, a knowledge of the equinox is not necessary.
The choice of the equinox, as a zero point of right as-
censions, is purely artificial, and a matter of convenience :
but as on the earth, any station (as a natioTial observa-
tory) may be chosen for an origin of longitudes ; so in
uranography, any conspicuous star may be selected as an
initial point from which hour angles may be reckoned,
and from which, by merely observing differences or in-
tervals of time, the situation of all others may be de-
duced. In practice, these intervals are aff'ected by cer-
tain minute causes of inequality, which must be allowed
for, and which will be explained in their proper places.
(246.) The declinations of celestial objects are ob-
tained, 1st, By observation of their meridiem altitudes ^
with the mural circle or other proper instruments. This
requires a knowledge of the geographical latitude of the
station of observation, which itself is only to be obtained
by celestial observation. 2dly, And more directly by ob-
servation of their polar distances on the mural circle,
as explained in art. 136, which is independent of any
previous determination of the latitude of the station ;
neither, however, in this case, does observation give
directly and immediately the exact declinations. The
observations require to be corrected, first for refraction,
and moreover for tliosc minute causes of inequality which
have been just alluded to in the case of right ascensions,
(247.) In this manner, then, may the places, one
among the other, of all celestial objects be ascertained,
and maps and globes constructed. Now here arises a
very important question. How far are these places per-
manent? Do these stars and the greater luminaries of
heaven preserve for ever one invariable connexion and
' relation of place inter se, as if they formed part of a
solid though invisible firmament; and, like the great
natural landmarks on the earth, preserve immutably the
same distances and bearings each from the other ? If so,
the most rational idea we could form of the universe
would be that of an earth at absolute rest in the centre,
an4 a hollow crystalline sphere circulating round it, aucl

154 A TREATISE ON ASTRONOMY. [CHAP. IV.

carrying sun, moon, and stars along in its tliurnal mo-
tion. If not, we must dismiss all such notions, and
inquire individually into the distinct history of each ob-
ject, with a view to discovering the laws of its peculiar
motions, and whether any and what other connexion
subsists between them.

(248.) So far is this, however, from being the case,
that observation, even of the most cursory nature, are
sufficient to show that some, at least, of the celestial
bodies, and those the most conspicuous, are in a state
of continual change of place among the rest. In the
case of the moon, indeed, the change is so rapid and re-
markable, that its alteration of situation with respect to
such bright stars as may happen to be near it, may be
noticed any fine night in a few hours ; and if noticed on
two successive nights, cannot fail to strike the most care-
less observer. With the sun, too, the change of place
among the stars is constant and rapid ; though, from the
invisibility of stars to the naked eye in the day-time, it
is not so readily recognised, and requires either the use
of telescopes and angular instruments to measure it, or
a lonsfer continuance of observation to be struck with it.
Nevertheless, it is only necessary to call to mind its
greater meridian altitude in summer than in winter, and
the fact that the stars which come into view at night
vary witii the season of the year, to perceive that a great
change must have taken place in that interval in its re-
lative situation with respect to all the stars. Besides the
sun and moon, too, there are several other bodies, called
planets, which, for the most part, appear to the naked
eye only as the largest and most brilliant stars, and which
offer the same phenomenon of a constant change of place
among the stars ; now approaching, and now receding
from, such of them as we may refer them to as marks ;
and, some in longer, some in shorter periods, making,
like the sun and moon, the complete tour of the heavens.

(249.) These, however, are exceptions to the general
rule. The innumerable mvdtitude of the stars which are
distributed over the vault of the heavens form a constel-
lation, which preserves, not only to the eye of the casual
observer, but to the nice examination of the astronomer,

CHAP. IV, 3 FIXED AND ERRATIC STARS. 155

a uniformity of aspect which, when contrasted Avith
the perpetual change in the configurations of the sun,
moon, and planets, may well be termed invariable.
It is not, intleed, that, by the refinement of exact mea-
surements prosecuted from age to age, some small
changes of apparent place, attributable to no illusion
and to no terrestrial cause, cannot be detected in some
of them ; — such are called, in astronomy, the proper
motions of the stars ; — but these are so excessively slow,
that their accumulated amount (even in those stars for
which they are greatest) has been insufficient, in the
whole duration of astronomical history, to produce any
obvious or material alteration in the appearance of the
starry heavens.

(250.) This circumstance, then, establishes a broad
distinction of the heavenly bodies into two great classes ;
' — the fixed, among which (unless in a course of obser-
vations continued for many years) no change of mutual
situation can be detected ; and the erratic, or wandering
— (which is implied in the word planet*) — including the
sun, moon, and planets, as well as the singidar class of
bodies termed comets, in whose apparent places among
the stars, and among each other, the observation of a few
days, or even hours, is sufficient to exhibit an indisputa-
ble alteration.

(251.) Uranography, then, as it concerns the fixed
celestial bodies (or, as they are usually called, the fixed
stars), is reduced to a simple marking down of their re-
lative places on a globe or on maps ; to the insertion on
that globe, in its due place in the great constellation of
the stars, of the pole of the heavens, or the vanishing
point of parallels to the earth's axis ; and of the equa-
tor and place of the equinox: points and circles these,
which though artificial, and having reference entirely to
our earth, and therefore subject to all changes (if any) to
which the earth's axis may be liable, arc yet so con-
venient in practice, that they have obtained an admission
(with some other circles and lines), sanctioned by usage,
in all globes and planispheres. The reader, however,
will take care to keep them separate in his mind, and to
* HAavnTHf , a wanderer.

156 A tREAtlSE ON A^tRONOMIr. [cHAP. iVi

familiarize himself with the idea rather of tivo or more
celestial globes, superposed and fitting on each other, on
one of which — a real one — are inscribed the stars ; on
the others those imaginary points, lines, and circles
which astronomers have devised for their own uses, and
to aid their calculations ; and to accustom himself to
Conceive in the latter. Or artificial, spheres a capability
of being shifted in any manner upon the surface of the
other ; so that, should experience demonstrate (as it
does) that these artificial points and lines are brought,
by a slow motion of the earth's axis, or by other seculat
variations (as they are called), to coincide, at very dis-
tant intervals of time, with different stars, he may not
be unprepared for the change, and have no confusion to
correct in his notions.

(252.) Of course we do not here speak of those un-
couth figures and outlines of men and monsters, which
are usually scribbled over celestial globes and maps, and
serve, in a rude and barbarous way, to enable us to talk
of groups of stars, or districts in the heavens, by names
which, though absurd or puerile in their origin, have
obtained a currency from which it would be difficult,
and perhaps wrong, to dislodge them. In so far as they
have really (as some have) any slight resemblance to the
figures called up in imagination by a view of the more
splendid " constellations," they have a certain conve*
nience ; but as they are otherwise entirely arbitrary, and
correspond to no natural subdivisions or groupings of
the stars, astronomers treat them lightly, of altogether
disregard them,* except for briefly naming remarkable
stars, as a. Leonis, ^ Scorpii, &c. &;c., by letters of the
Greek alphabet attached to them. The reader will find
them on any celestial charts or globes, and may compare
them with the heavens, and there learn for himself their
position.

*This disregard is neither supercilious nor causeless. The constella'
tions seem to have been almost purposely named and delineated to cause
as much confusion and inconvenience as possible. Innumerable snakes
twine through long and contorted areas of the heavens, where no me-
mory can follow them; bears, lions and fishes, large and small, northern
and southern, confuse all nomenclature, &c. A better system of con--
stellations might have been a material help t*s an artificial memory.

CHAP. IV.] THE MILKY WAY. — THE ZODIAC. 157

(253.) There are not wanting, however, natural dis-
tricts in the heavens, which ofi'er great peculiarities of
character, and strike every observer: such is the milky
way, that great luminous band, which stretches, every
evening, all across the sky, from horizon to horizon,
and wlaich, when traced with diligence, and mapped
down, is found to form a zone completely encircling the
whole sphere, almost in a great circle, which is neither
an hour circle, nor coincident with any other of our
astronomical grammata. It is divided in one part of its
course, sending off a kind of branch, which unites again
with the main body, after remaining distinct for about
150 degrees. This remarkable belt has maintained,
from the earliest ages, the same relative situation among
the stars ; and, when examined through powerful tele-
scopes, is found (wonderful to relate !) to consist entirely
of stars scattered by millions, like glittering dust, on
the black ground of the general heavens.

(254.) Another remarkable region in the heavens ia
the zodiac, not from any thing peculiar in its own con-
stitution, but from its being the area within which the
apparent motions of the sun, moon, and all the gi-eater
planets are confined. To trace the path of any one of these,
it is only necessary to ascertain, by continued observa-
tion, its places at successive epochs, and entering these
upon our map or sphere in sufficient number to form a
series, not too far disjoined, to connect them by lines
from point to point, as we mark out the course of a ves-
sel at sea by mapping down its place from day to day.
Now when this is done, it is found, first, that the appa-
rent path, or track, of the sun on the surface of the hea-
vens, is no other than an exact great circle of the sphere
which is called the ecliptic, and which is inclined to the
equinoctial at an angle of about 23° 28', intersecting it at
two opposite points, called the equinoctial points, or
equinoxes, and which are distinguished from each other
by the epithets vernal and autumnal ; the vernal being
that at which the sun crosses the equinoctial from south
to north ; the autumnal, when it quits the northern and
enters the southern hemisphere. Secondly, that the
moon and all the planets pursue paths- which, in like

O

158 A TREATISE ON ASTRONOMY. [cHAP. IV.

manner, encircle the whole heavens, but are not, like
that of the sun, great circles exactly returning into them-
selves and bisecting the sphere, but rather spiral curves
of mucli complexity, and described with very unequal
velocities in their different parts. They have all, how-
ever, this in common, tliat the general direction of their
motions is the same with that of the sun, viz. from tvest
to east, that is to say, the contrary to that in whicli both
they and the stars appear to be carried by the diurnal
motion of the heavens ; and, moreover, that they never
deviate far from the ecliptic on either side, crossing and
recrossing it at regular and equal intervals of time, and
confining f.iemselves within a zone, or belt (the zodiac
already spoken of), extending 9° on either side of the
ecliptic. ,

(255.) It would manifestly be useless to map down on
globes or charts the apparent paths of any of those bodies
which never retrace the same course, and Avhich, there-
fore, demonstrably, must occupy at some one moment
or other of their history, every point in the area of that
zone of the heavens within wliich they are circum-
scribed. The apparent complication of their movements
arises (that of the moon excepted) from our viewing
them from a station which is itself in motion, and would
disappear, could we shift our point of view and observe
them from the sun. On the other hand, the apparent
motion of the sun is presented to us under its least in-
volved form, and is studied, from the station we occupy,
to the greatest advantage. So that, independent of the
importance of that luminary to us in other respects, it is
by the investigation of tlie laws of its motions in the first
instance that we must rise to a knowledge of those of all
the other bodies of our system.

(256.) The ecliptic, which is its apparent path among
the stars, is traversed by it in the period called the side-
real year, which consists of 365"* 6" 9" 9'- 6, reckoned
in mean sqlar time, or 366'* 6^ 9"" 9'*6, reckoned in si-
dereal time. The reason of this difference (and it is this
which constitutes the origin of the difference between
solar and sidereal time) is, that as the sun's apparent
annual motion among the stars is performed in a con-

CHAP. IV.] THE ECLIPTIC. SIDEREAL YEAR. 159

trary direction to the apparent diurnal motion of both
sun and stars, it comes to the same thing as if the diur-
nal motion of the siui were so much slower than that of
the stars, or as if the sun lagged behind them in its
daily course. Where this has gone on for a whole year,
the sun will have fallen behind the stars by a whole
circumference of the heavens — or, in other words — in a
year, the sun will have made fewer diurnal revolutions,
by one, than the stars. So that the same interval of time
which is measured by 366'' 6*", &c. of sidereal time, if
reckoned in mean solar days, hours, &c. will be called
SeS'' &^, Sic. Tims, then, is the proportion between
the mean solar and sidereal day established, which,
reduced into a decimal fraction, is that of 1-00273791 to
1. The measurement of time by these different stand-
ards may be compared to that of space by the standard
feet, or ells of two different nations ; the proportion of
which, once settled, can never become a source of error.

(257.) The position of the ecliptic among the stars
may, for our present purpose, be regarded as invariable.
It is true that this is not strictly the case ; and on com-
paring together its position at present with that which
it held at the most distant epoch at which we possess
observations, we find evidences of a small change, which
theory accounts for, and whose nature will be hereafter
explained ; but that change is so excessively slow, that
for a great many successive years, or even for whole
centuries, this circle may be regarded as holding the
same position in the sidereal heavens.

(258.) The poles of the ecliptic, like those of any
other great circle of the sphere, are opposite points on
its surface, equidistant from the ecliptic in every direc-
tion. They are of course not coincident with those of
the equinoctial, but removed from it by an angular in-
terval equal to the inclination of the ecliptic to the equi-
noctial (23° 28'), which is called the obliqiuty of the
ecliptic. In the annexed figure, if Pja represent the north
and south poles (by which, when used without qualifi-
cation we always mean the poles of the equinoctial),
and EQAV the equinoctial, VSAW the ecliptic, and Kk,
its poles — the spherical angle QVS is the obliquity of the

160 A TREATISE ON ASTRONOMY. [cHAP. IV.

ecliptic, and is equal in angular measure to PK or SQ.
If we suppose the sun's apparent motion to be in the
direction VSAW, V will be the vernal and A the uu-
tumnal equinox. S and W, the two points at which
the ecliptic is most distant from the equinoctial, are
termed solstices, because, when arrived there, the sun
ceases to recede from the equator, and (in that sense, so
far as its motion in declination is concerned) to stand
still in the heavens. S, the point Avhere the sun has the
greatest northern declination, is called the summer sol-
stice, and W, that where it is farthest south, the tvinter.
These epithets obviously have their origin in the depend-
ence of the seasons on the sun's declination, which will
be explained in the next chapter. The circle EKPQA;j3,
which passes through the poles of the ecliptic and equinoc-
tial, is called the solstitial colure ; and a meridian drawn
through the equinoxes, PV/jA, the equinoctial colure.

(259.) Since the ecliptic holds a determinate situation
in the starry heavens, it may be employed, like the equi-
noctial, to refer the positions of the stars to, by circles
drawn through them from its poles, and therefore per-
pendicular to it. Such circles are termed, in astronomy,
circles of latitude — the distance of a star from the eclip-
tic, reckoned on the circle of latitude passing through it,
is called the latitude of the stars — and the arc of the

ecliptic intercepted between the vernal equinox and this
circle, its longitude. In the figure X is a star, PXR a

CHAP. IV.] CELESTIAL LONGITUDES AND LATITUDES. 161

circle of declination drawn throiigli it, by which it is
referred to the equinoctial, and KXT a circle of latitude
referring it to the ecliptic— then, an VR is the right
ascension, and RX the declination, of X, so also is VT
its longitude, and TX its latitude. The use of the terms
longitude and latitude, in this sense, seems to have ori-
ginated in considering the ecliptic as forming a kind of
natural equator to the heavens, as the terrestrial equator
does to the earth — the former holding an invariable po-
sition with respect to the stars, as the latter does with
respect to stations on the earth's surtace. The force of
this observation will presently become apparent.

(260.) Knowing the right ascension and declination of
an object, we niay find its longitude and latitude, and vice
versa. This is a problem of great use in physical astro-
nomy. The following is its solution : In our last figure,
EKPQ, the solstitial colure is of course 90° distant
from V, the vernal equinox, which is one of its poles —
so that VR (the right ascension) being given, and also
VE, the arc ER, and its measure, the spherical angle
EPR, or KPX, is known. In the spherical triangle
KPX, then we have given, 1st, The side P K, which,
being the distance of the poles of the ecliptic and equi-
noctial, is equal to the obliquity of the ecliptic ; 2d, The
side PX, the polar distance, or the complement of the
declination RX ; and 3d, the included angle KPX ; and
therefore, by spherical trigonometry, it is eas)' to find the
other side KX, and the remaining angles. Now KX is
the complement of the required latitude XT, and the
angle PKX being known, and PKV being a right
angle (because SV is 90°), the angle XKV becomes
known. Now this is no other than the measure of the
longitude VT of the object. The inverse problem is
resolved by the same triangle, and by a process exactly
similar.

(261.) The same course of observations by which the
path of the sun among the fixed stars is traced, and the
ecliptic marked out among them, determines, of course,
the place of the equinox V upon the starry sphere, at
that time — a point of great importance in practical astro-
nomy, as it is the origin or zero point of right ascension

o2

162 A TREATISE ON ASTRONOMY. [cHAP. IV.

Now, when this process is repeated at considerably dis-
tant intervals of time, a very remarkable phenomenon is
observed ; viz. that the equinox does not preserve a con-
stant place among the stars, but shifts its position, travel--
ling continually and regularly, although with extreme
slowness, backwards, along the ecliptic, in the direction
VW from east to west, or the contrary to that in which the
sun appears to move in that circle. The equinoctial point
thus moving, as it were, to meet the sun in his apparent an-
nual round, the sun arrives at the equinoctial point sooner ;
that is, the time of the equinox happens sooner than
it would otherwise do : hence the recession of the equi-
noctial point causes a /jrecesston in the time of the equinox.
The amount of this motion by which the equinox travels
backward, or retrogrades (as it is called), on the ecliptic,
is 0° 0' 50"* 10 per annum, an extremely minute quan-
tity, but which, by its continual accumulation from year
to year, at last makes itself very palpable, and that in a
way highly inconvenient to practical astronomers, by
destroying, in the lapse of a moderate number of years,
the arrangement of their catalogues of stars, and making
it necessary to reconstruct them. Since the formation
of the earliest catalogue on record, the place of the equi-
nox has retrograded already about 30°. The period in
which it performs a complete tour of the ecliptic, is
25,868 years.

(262.) The immediate uranographical effect of the
precession of the equinoxes is to produce a uniform in-
crease of longitude in all the heavenly bodies, whether
fixed or erratic. For the vernal equinox being the initial
point of longitudes, as well as of right ascension, a re-
treat of this point on the ecliptic tells upon the longi-
tudes of all alike, whether at rest or in motion, and pro-
duces, so far as its amount extends, the appearance of a
motion in longitude common to all, as if the whole hea-
vens had a slow rotation round the poles of the ecliptic
in the long period above mentioned, similar to what they
have in twenty-four hours round those of the equinoctial.

(263.) To form a just idea of this curious astronomi-
cal phenomenon, however, we must abandon, for a time,
the consideration of the ecliptic, as tending to produce

CHAP. IV. J PRECESSION OF THE EQUINOXES. 163

confusion in our ideas ; for this reason, that the stability
of the ecliptic itself among the stars is (as already hinted,
art. 257) only approximate, and that in consequence its
intersection with the equinoctial is liable to a certain
amount of change, arising from its fluctuation, which
mixes itself with what is due to the principal uranogra-
phical cause of the phenomenon. This cause will be-
come at once apparent, if, instead of regarding the equi-
nox, we fix our attention on the pole of the equinoc-
tial, or the vanishing point of the earth's axis.

(264.) The place of this point among the stars is easily
determined, at any epoch, by the most direct of all astro-
nomical observations, — those with the mural circle. By
this instrument we are enabled to ascertain at every mo-
ment the exact distance of the polar point from any
three or more stars, and therefore to lay it down, by
triangulating from these stars, with unerring precision,
on a chart or globe, without the least reference to the
position of the ecliptic, or to any other circle not natu-
rally connected with it. Now, when this is done with
proper diligence and exactness, it results that, although
for short intervals of time, such as a few days, the place
of the pole may be regarded as not sensibly variable, yet
in reality it is in a state of constant, although extremely
slow motion ; and, what is still more remarkable, this
motion is not uniform, but compounded of one principal,
uniform, or nearly uniform, part, and other smaller and
subordinate periodical fluctuations : the former giving
rise to the phenomena of jurecession; the latter to another
distinct phenomenon called nutation. These' two phe-
nomena, it is true, belong, theoretically speaking, to one
and the same general head, and are intimately connected
together, forming part of a great and complicated chain
of consequences flowing from the earth's rotation on its
axis : but it will be of advantage to present clearness to
consider them separately.

(265.) It is found, then, that in virtue of the uniform
part of the motion of the pole, it describes a circle in the
heavens around the pole of the ecliptic as a centre, keep-
ing constantly at the same distance of 23^ 28' from it,
in a direction from east to west, and with such a velocity,

164 A TREATISE ON ASTRONOMY. [cHAP. IV.

that the annual angle described by it, in this its imaginary
orbit, is 50"-10; so that the whole circle would be de-
scribed by it in the above-mentioned period of 25,868
years. It is easy to perceive how such a motion of the
pole will give rise to the retrograde motion of the equi-
noxes ; for in the figure, art. 259, suppose the pole P in
the progress of its motion in the small circle POZ round
K to come to O, then, as the sitviation of the equinoctial
EVQ is determined by tliat of the pole, this, it is evi-
dent, must cause a displacement of the equinoctial, which
Avill take a new situation, EUQ, 90° distant in every
part from the new position O of the pole. The point U,
therefore, in which the displaced equinoctial will inter-
sect the ecliptic, i. e. the displaced equinox, will lie on
that side of V, its original position, towards which the
motion of tlje pole is directed, or to the westward.

(266.) The precession of the equinoxes thus conceived,
consists, then, in a real but very slow motion of the pole
of the heavens among the stars, in a small circle round
the pole of the ecliptic. Now this cannot happen with-
out producing corresponding changes in the apparent
diurnal motion of the sphere, and the aspect which the
heavens must present at very remote periods of history.
The pole is nothing more than the vanishing point of the
earth's axis. As this point, then, has such a motion as
described, it necessarily follows that the earth's axis must
have a conical motion, in virtue of which it points suc-
cessively to every part of the small circle in question.
We may form the best idea of such a motion by noticing
a child's peg-top, when it spins not upright, or that amus-
ing toy the te-to-tum, which, when delicately executed,
and nicely balanced, becomes an elegant philosophical
instrument, and exhibits, in the most beautiful manner,
the whole phenomenon, in a way calculated to give at
once a clear conception of it as a fact, and a considerable
insight into its physical cause as a dynamical effect. The
reader will take care not to confound the variation of the
position of the earth'' s axis in space with a mere shifting
of the imaginary line about which it revolves, in its inte-
rior. The whole earth participates in the motion, and
goes along with the axis as if it were really a bar of iron

CHAP. IV.] NUTATION. 165

driven through it. That such is the case is proved by
the two great facts: 1st, that tlie latitudes of places on
the earth, or their geographical situation with respect to
the poles, have undergone no pei'ceptible change from
the earliest ages. 2dly, that the sea maintains its level,
which could not be the case if the motion of the axis
were not accompanied with a motion of the whole mass
of the earth.

(267.) The visible effect of precession on the aspect
of the heavens consists in the apparent approach of
some stars and constellations to the pole and recess of
others. The bright star of the Lesser Bear, which we
call the pole star, has not always been, nor will always
continue to be, our cynosure : at the time of the con-
struction of the earliest catalogues it was 12° from the
pole — it is now only 1° 24', and will approach yet nearer,
to within half a degree, after which it will again recede,
and slowly give place to others, which will succeed it in
its companionship to the pole. After a lapse of about
12,000 years, the star a. Lyrre, the brightest in the north-
ern hemisphere, will occupy the remarkable situation of
a pole star, approaching within about 5° of the pole.

(268.) The nutation of the earth's axis is a small and
slow subordinate gyratory movement, by which, if sub-
sisting alone, the pole would describe among the stars,
in a period of about nineteen years, a minute ellipsis,
having its longer axis equal to 18"'5, and its shorter to
13"*74 ; the longer being directed towards the pole of
the ecliptic, and the shorter, of course, at right angles to
it'. The consequence of this real motion of the pole is
an apparent approach and recess of all the stars in the
heavens to the pole in the same period. Since, also, the
place of the equinox on the ecliptic is determined by the
place of the pole in the heavens, the same cause will
give rise to a small alternate advance and recess of the
equinoctial points, by which, in the same period, both
the longitudes and the right ascensions of the stars will
be also alternately increased and diminished.

(269.) Both these motions, however, although here
considered separately, subsist jointly ; and since, while
in virtue of tlie nutation, the pole is describing its little

166 A TREATISE ON ASTRONOMY. [cHAP. IV.

ellipse of 18"*5 in diameter, it is carried by the gi-eater
and regularly progressive motion of precession over so
much of its circle round the pole of the ecliptic as cor-
responds to nineteen years, — that is to say, over an angle
of nineteen times 50"- 1 round the centre (which, in a
small circle of 23° 28' in diameter, corresponds to 6' 20",
as seen from the centre of the sphere) : the path which
it will pursue in virtue of the tAvo motions, subsisting
jointly, will be neither an ellipse nor an exact circle, but
a gently nndulated ring like that in tlie iigure (where,
however, the undulations are much exaggerated). (See
fig. to art. 272.)

(270.) These movements of precession and nutation
are common to all the celestial bodies both fixed and er-
ratic ; and this circumstance makes it impossible to attri-
bute them to any other cause than a real motion of the
earth's axis, such as we have described. Did they only
affect the stars, they might, with equal plausibility, be
urged to arise from a real rotation of the starry heavens,
as a solid shell round an axis passing through the poles
of the ecliptic in 25,868 years, and a real elliptic gyration
of that axis in nineteen years : but since they also affect
the sun, moon, and planets, which, having motions inde-
pendent of the general body of tlie stars, cannot without ex-
travagance be supposed attached to the celestial concave,*
this idea falls to the ground ; and there only remains,
then, a real motion in the earth by wliich they can be
accounted for. It will be shown in a subsequent chapter
that they are necessary consequences of the rotation of
the earth, combined with its elliptical figure, and the un-
equal attraction of the sun and moon on its polar and
equatorial regions.

(271.) Uranographically considered, as affecting the
apparent places of the stars, they are of the utmost im-
portance in practical astronomy. When we speak of the
right ascension and declination of a celestial object, it
becomes necessary to state what epoch we intend, and

* This argument, cogent as it is, acquires additional and decisive force
from the law of nutation, which is dependent on the position, for the time,
of the lunar orbit. If we attribute it to a real motion of the celestial
(Sphere, we must then maintain that splicre to bo kept in a constant state
oi tremor by the motion of the moon I

EQUATIONS FOR PRECESSION AND NUTATION. 167

whether we ineau t!ie mean right ascension ; cleared, that
is, of the periodical fluctuation in its amount, which
arises from nutation, or the apparent right ascension,
which being reckoned from the actual place of the vernal
equinox, is affected by the periodical advance and recess
of the equinoctial point thence produced — and so of the
other elements. It is the practice of astronomers to re-
duce, as it is termed, all their observations, both of right
ascension and declination, to some common and conve-
nient epoch — such as the beginning of the year for tem-
porary purposes, or of the decade, or the century for
more permanent uses, by subti-acting from them the
whole effect of precession in the interval ; and, moreover,
to divest them of the influence of nutation by investiga-
ting and subducting the amount of change, both in right
ascension and declination, due to the displacement of the
pole from the centre to the circumference of the little el-
lipse above mentioned. This last process is technically
termed correcting or equating the observation for nuta-
tion; by which latter word is always understood, in as-
tronomy, the getting rid of a periodical cause of fluctua-
tion, and presenting a result, not as it was observed, but
as it would have been observed, had that cause of fluc-
tuation had no existence.

(272.) For these purposes, in the present case, very
convenient formulae have been derived, and tables con-
structed. They are, however, of too technical a charac-
ter for this work ; we shall, however, point out the man-
ner in which the investigation is conducted. It has been
shown in art. 260 by what means the right ascension and
declination of an oljject are derived from its longitude
and latitude. Referring to the figure of that article, and
supposing the triangle KPX orthographically projected
on the plane of the ecliptic as in the annexed figure : in
the triangle KPX, KP is the obliquity of the ecliptic,
KX the co-latitude (or complement of latitude), and the
angle PKX the co-longilude of the object X. These
are the data of our question, of which the first is con-
stant, and the two latter are varied by the effect of pre-
cession and nutation ; and their variations (considering
the minuteness of the latter efiect generally, and the

168 A TREATISE ON ASTRONOMY. £cHAP. IV.

small number of years in comparison of the whole period
of 25,868, for which we ever require to estimate the
effect of the former) are of that order which may be
regarded as infinitesimal in geometry, and treated as such
without fear of error. The whole question, then, is re-
duced to this : — In a spherical triangle KPX, in which
one side KX is constant, and an angle K, and adjacent

side KP vary by given infinitesimal changes of the po-
sition of P : required the changes thence arising in the
other side PX, and the angle KPX? This is a very
simple and easy problem of spherical geometry, and be-
ing resolved, it gives at once the reductions we are seek-
ing ; for PX being the polar distance of the object, and
the angle KPX its right ascension plus 90°, their va-
riations are the very quantities we seek. It only re-
mains, then, to express in proper form the amount of the
precession and nutation in longitude and latitude, when
their amount in right ascension and declination will im-
mediately be obtained.

(273.) The precession in latitude is zero, since the
latitudes of objects are not changed by it : that in lon-
gitude is a quantity proportional to the time at the rate
of 50"*10 per annum. With regard to the nutation in
longitude and latitude, these are no other than the ab-
scissa and ordinate of the little ellipse in which the pole

CHAP. IV.J ABERRATION OF LIGHT. 160

moves. The law of its motion, however, therein, cannot
be understood till the reader has been made acquainted
with the principal features of the moon's motion on
Avhich it depends. See chap. XI.

(274.) Another consequence of what has been shown
respecting precession and nutation is, that sidereal time
as astronomers use it, i. e. as reckoned from the transit
of the equinoctial point, is, not a mean or uniformly
Jlotving quantity, being affected by nutation ; and,
moreover, tliat so reckoned, even when cleared of the
periodical lluctuation of nutation, it does not strictly
correspond to the earth's diurnal rotation. As the sun
loses one day in the year on the stars, by its direct mo-
tion in longitude ; so the equinox gains one day in
25,868 years on them by its retro gradation. We ought,
therefore, as carefully to distinguish between mean and
apparent sidereal as between mean and apparent solar
time.

(275.) Neither precession nor nutation change the
apparent places of celestial objects inter se. We see
them, so far as these causes go, as they are, though from
a station more or less unstable, as we see distant land
objects correctly formed, though appearing to rise and
fall when viewed from the heaving deck of a ship in the
act of pitching and rolling. But there is an optical cause,
independent of refraction or of perspective, which dis-
places them one among the other, and causes us to view
the heavens under an aspect always to a certain slight
extent false ; and whose influence must be estimated and
allowed for before we can obtain a precise knowledge of
the place of any object. This cause is what is called
the aberration of light ; a singular and surprising effect
arising from this, that we occupy a station not at rest
but in rapid motion ; and that the apparent directions of
the rays of light are not the same to a spectator in mo-
tion as to one at rest. As the estimation of its effect be-
longs to uranography, we must explain it here, though,
in so doing, we must anticipate some of the results to be
detailed in subsequent chapters.

(276.) Suppose a shower of rain to fall perpendicularly
in a dead calm ; c person exposed to the shower, who

P

170 A TREATISE ON ASTRONOMY. [cHAP. IV.

should stand quite still and upright, would receive the
drops on his hat, which would thus shelter him, but if
he ran forward in any direction they would strike him in
the face. The efleet would be the same as if he remained
still, and a wijid should arise of the same velocity, and
drift them against him. Suppose a ball let fall from a
point A above a horizontal line EF, and that at B were
placed to receive it the open mouth of an inclined hollow

QA.

tube PQ ; if the tube were held immoveable, the ball
would strike on its lower side, but if the tube were car-
ried forward in the direction EF, with a velocity properly
adjusted at every instant to that of the ball, while pre-
serving its inclination to the horizon, so that when the
ball in its natural descent reached C, the tube sliould
have been carried into the position RS, it is evident that
the ball would, throughout its whole descent, be found
in the axis of the tube ; and a spectator, referring to tlie
tube the motion of the ball, and carried along with the
former, unconscious of its motion, wovdd fancy that the
ball had been moving in the inclined direction RS of the
tube's axis.

(277.) Our eyes and telescopes are such tubes. In
whatever manner we consider light, whether as an ad-
vancing wave in a motionless ether, or a shower of
atoms traversing space, if in the interval between the
rays traversing the object-glass of the one or the coraea
of the other {at ivhich moment they acquire that con-
vergence which directs them to a certain point in fixed

CHAP. IV.] CORRECTION FOR ABERRATION. 171

space), the cross wires of the one or the retina of the
Other be slipped aside, tlie point of convergence (which
remains unchanged) will no longer correspond to the in-
tersection of the wires or tlie central point of our visual
area. The object then will appear displaced ; and the
amount of this displacement is aberration.

(278.) The- earth is moving through space with a ve-
locity of about 19 miles per second, in an elliptic path
round the sun, and is therefore changing the direction
of its motion at every instant. Light travels with a ve-
locity of 192,000 miles per second, which, although
much greater than tliat of the earth, is yet not infinitely
so. Time is occupied by it in traversing any space, and
in that time the earth describes a space which is to the
former as 19 to 192,000, or as the tangent of 20"'5 to
radius. Suppose now APS to represent a ray of light
from a star at A, and let the tube PQ be tliat of a tele-
scope so inclined forward that the focus formed by its
object-glass shall be received upon its cross wire, it is
evident from what has been said, that the inclination of
the tube must be such as to make PS : SQ : : velocity of
light : velocity of the earth, : : tan. 20"*5 : 1 ; and,
therefore, the angle SPQ, or PSR, by which the axis of
the telescope must deviate from the true direction of the
star, must be 20 ""5.

(279.) A similar reasoning will hold good when the
direction of the earth's motion is not perpendicular to
the visual ray. If SB be the
true direction of the visual
ray, and AC the position in
which the telescope requires
to be held in the apparent di-
rection, we must still have tlie
proportion BC : BA : : velo- Ji iV>

locity of light : velocity of the earth : : rad. : sine of 20"*5
(for in such small angles it matters not whether we use
the sines or tangents). But we have, also, by trigono-
metry, BC : BA : : sine of BAC : sine of ACB or CBD,
which last is the apparent displacement caused by aber-
ration. Thus it appears that the sign of the aberration, or
(since the angle is extremely small) tlie aberration itself,

173 A TREATISE ON ASTR0N0M7. [[ciIAP. IV.

is proportional to the sine of the angle made by the earth's
motion in space with the visual ray, and is therefore a
maximum when the line of sight is perpendicular to the
direction of the earth's motion.

(280.) The uranographical effect of aberration, then,
is to distort the aspect of the heavens, causing all the
stars to crowd, as it were, directly towards that point in
the heavens which is the vanishing point of all lines
parallel to that in which the earth is for the moment
moving. As the earth moves round the sun in the plane
of the ecliptic, this point must lie in that plane, 90° in
advance of the earth's longitude, or 90° behind the sun's,
and shifts of course continually, describing the circum-
ference of the ecliptic in a year. It is easy to demon-
strate that the effect on each particular star will be to
make it apparently describe a small ellipse in the heavens,
having for its centre the point in which the star would
be seen if the earth were at rest.

(281.) Aberration then affects the apparent right as-
censions and declinations of all the stars, and that by
quantities easily calculable. The formulae most conve-
nient for that purpose, and which, systematically embrac-
ing at the same time the corrections for precession and
nutation, enable the observer, with the utmost readiness,
to disencumber his observations of right ascension and
declination of their influence, have been constructed by
Prof. Bessel, and tabulated in the appendix to the first
volume of the Transactions of the Astronomical Society,
where they will be found accompanied with an extensive
catalogue of the places, for 1830, of the principal fixed
stars, one of the most useful and best arranged works
of the kind which has ever appeared.

(282.) When the body from which the visual ray
emanates is, itself, in motion, the best way of conceiving
the effect of aberration (independently of theoretical
views respecting the nature of light)* is as follows. The

* The results of the undulatory and corpuscular theories of light, in
the matter of aberration, are, in the main, the same. We say in the main.
There is, however, a minute difii^rence even of numerical results. In
the undulatory doctrine, the propagation of light takes place with equal
velocity in all directions whether the luminary be at rest or in motion.
bi the corpuscular, with an excess of velocity in the direction of the

CHAP. IV.3 URANOGRAPHICAL PROBLEMS. 173

ray by which we see any object is not that which it emits
at the moment we look at it, but that which it did emit
some time before, viz. the time occupied by light in tra-
versing the interval which separates it from us. The
aberration of such a body then arising from the earth's
velocity must be applied as a correction, not to the line
joining the earth's place at the moment of observation
with that occupied by the body at the same moment,
but at that antecedent instant when the ray quitted it.
Hence it is easy to derive the rule given by astronomical
writers for the case of a moving object. From the known
laws of its motion and the earth'' s, calcidate its apparent
or relative angular motion in the time taken by light to
traverse its distance from the earth. This is its aberra-
tion, and its effect is to displace it in a direction contrary
to its apparent relative motion among the stars.

We shall conclude this chapter with a few uranogra-
phical problems of frequent practical occurrence, which
may be resolved by the rules of spherical trigonometry.

(283.) Of the following five quantities, given any three,
to find one or both the others.

1st, The latitude of the place ; 2d, the declination of an
object ; 3d, its hour angle east or west from the meridian ;
4th, its altitude ; 5th, its azimuth.

In tlie figure of art. 94, P is the pole, Z the zenith, and
S the star ; and the five quantities above mentioned, or
their complements, constitute the sides and angles of the
spherical triangle PZS ; PZ being the co-latitude, PS
the co-declination or polar distance ; SPZ the hour an-
gle Ti-^S the co-altitude or zenith distance ; and PZS the
azimuth. By the solution of this spherical triangle, then,
all problems involving the relations between these quanti-
ties may be resolved.

(284.) For example, suppose the time of rising or set-
ting of the sun or of a star were required, having given
its right ascension and polar distance. The star rises

motion over that in the contrary equal to twice the velocity of the body's
motion. In the cases, then, of a body moving with equal velocity directly
to and directly from the earth, the aberrations will be alike on the undu-
latory, but different on the corpuscular hypothesis. The utmost difier-
ence which can arise from this cause in our s^yslem cannot amount to
above six thousandths of a second.

p2

174 A TREATISE ON ASTROMOMY. [cHAP. IV.

when apparently on the horizon, or really about 34' be-
low it (owing to refraction), so that, at the moment of its
apparent rising, its zenith distance is 90° 34'=ZS. Its
polar distance PS being also given, and the co-latitude ZP
of the place, we have given the three sides of the trian-
gle, to find the hour angle ZPS, which, being known, is
to be added to or subtracted from the star's right ascen-
sion, to give the sidereal time of setting or rising, which,
if we please, may be converted into solar time by the
proper rules and tables.

(285.) As another example of the same triangle, we
may propose to find the local sidereal time, and the lati-
tude of the place of observation, by observing equal
altitudes of the same star east and west of the meri-
dian, and noting the interval of the observations in side-
real time.

The hour angles corresponding to equal altitudes of a
fixed star being equal, the hour angle east or west will be
measured by half the observed interval of the observa-
tions. In our triangle, then, we have given this hour an-
gle ZPS, the polar distance PS of the star, and ZS, its
co-altitude at the moment of observation. Hence we may
find PZ, the co-latitude of the place. Moreover, the
hour angle of the star being known, and also its right as-
cension, the point of the equinoctial is known, which is
on the meridian at the moment of observation ; and,
therefore, the local sidereal time at that moment. This
is a Vtry useful observation for determining the latitude
and time at an unknown station.

(286.) It is often of use to know the situation of the
ecliptic in the visible heavens at any instant ; that is to
say, the points where it cuts the horizon, and the altitude
of its highest point, or, as it is sometimes called, the
nonagesimal point of the ecliptic, as well as the longitude
of this point on the ecliptic itself from the equinox.
These, and all questions referable to the same data andquae-
sita, are resolved by the spherical triangle ZPE, formed
by the zenith Z (considered as the pole of the horizon),
the pole of the equinoctial P, and the pole of the ecliptic
E. The sidereal time being given, and also the right
ascension of the pole of the ecliptic (which is always the

CHAP. IV.] ITRANOGRAPHICAL PROBLEMS. 175

z

same, viz. 18'' C" 0'), the hour angle ZPE of that point
is known. Then, in this triangle we have given PZ, the
co-latitude ; PE, the polar distance of the pole of the
ecliptic, 23° 28', and the angle ZPE ; from which we
may find, 1st, the side ZE, which is easily seen to be
equal to the altitude of the nonagesimal point sought ;
and, 2dly, the angle PZE, which is the azimuth of the
pole of the ecliptic, and which, therefore, being added to
and subtracted from 90°, gives the azimuths of the eastern
and western intersections of the ecliptic with the horizon.
Lastly, the longitude of the nonagesimal point may be
had, by calculating in the same triangle the angle PEZ,
which is its complement,

(287.) The angle of situation oi ^ star is the angle in-
cluded at the star between circles of latitude and of decli-
nation passing through it. To determine it in any pro-
posed case, we must resolve the triangle PSE, in which
are given PS, PE, and the angle SPE, which is the dif-
ference between the star's right ascension and 18 hours ;
from which it is easy to find the angle PSE required.
This angle is of use in many inquiries in physical astro-
nomy. It is called in most books on astronomy the an-
gle of position ; but the latter expression has become
otherwise, and more conveniently, appropriated.

(288.) From these instances, the manner of treating
sucli questions in uranography as depend on spherical
trigonometry will be evident, and will, for the most part,
offer little difficulty, if the student will bear in mind, as a

176 ■ A TREATISE ON ASTRONOMY. [cHAP. V.

practical maxim, rather to consider the poles of the great
circles which his question refers to, than the circles
themselves.

CHAPTER V.

OF THE sun's motion.

Apparent Motion of the Sun not uniform — Its apparent Diameter also va-
riable— Variation of its Distance concluded — lis apparent Orbit an El-
lipse about the Focus — Law of the angular Velocity — Equable Descrip-
tion of Areas — Parallax of the Sun — lis Distance and Magnitude —
Copernican Explanation of the Sun's apparent Motion — Parallelism of
the Earth's Axis — The Seasons — Heat received from the Sun in differ-
ent Parts of the Orbit.

(289.) In the foreg-oing chapters, it has been shown
that the apparent path of the sun is a great circle of the
sphere, which it performs in a period of one sidereal
year. From this it follows, that the line joining the
earth and sun lies constantlj^ in one plane; and that,
therefore, whatever be the real motion from which this
apparent motion arises, it must be confined to one plane,
which is called the plane of the ecliptic.

(290.) We have already seen (art. 118) that the sun's
motion in right ascension among the stars is not uniform.
This is partly accounted for by the obliquity of the eclip-
tic, in consequence of which equal variations in longitude
do not correspond to^equal changes of right ascension.
But if we observe the place of the sun daily throughout
the year, by the transit and circle, and from these calcu-
late the longitude for each day, it will still be found that,
even in its own proper path, its apparent angular motion
is far from uniform. The change of longitude in twenty-
four mean solar hours averages 0° 59' 8"-33 ; but about
the 31st of December it amounts to 1° 1' 9"'9, and about
the 1st of July is only 0° 57' 11 "'5. Such are the ex-
treme limits, and such the mean value of the sun's appa-
rent angular velocity in its annual orbit.

(291.) This variation of its angular velocity is accom-
panied Avith a corresponding change of its distance from
us. Tlie change of distance is recognised by a variation

CHAP. V.J FORM OF THE SOLAR ORBIT. 177

observed to take place in its apparent diameter, Avhen
measured at different seasons of the year, with an instru-
ment adapted for that purpose, called a heliometer,* or,
by calculating from the time which its disk takes to tra-
verse the meridian in the transit instrument. The great-
est apparent diameter corresponds to the 31st of Decern
ber, or to the greatest angular velocity, and measures 32'
35"-6 ; the least is 31' 31"-0, and corresponds to the 1st
of July ; at which epochs, as we have seen, the angular
motion is also at its extreme limit either way. Now, as
we cannot suppose the sun to alter its real size periodi-
cally, the observed change of its apparent size can only
arise from an actual change of distance. And the sines
or tangents of such small arcs being proportional to the
arcs themselves, its distances from us, at the above-named
epoch, must be in the inverse proportion of the apparent
diameters. It appears, therefore, that the greatest, the
mean, and the least distances of the sun from us are in
the respective proportions of the numbers 1*0 1679,
1-00000, and 0-98321 ; and that its apparent angular ve-
locity diminishes as the distance increases, and vice versa.
(292.) It follows from this, that the real orbit of the
sun, as referred to the earth supposed at rest, is not a
circle with the earth in the centre. The situation of the
earth within it is eccentric, the eccentricity amounting to

0*01679 of the mean distance, which may be regarded as
our unit of measure in this inquiry. But besides this,
the/on?i of the orbit is not circular, but elliptic. If from
any point O, taken to represent the earth, we draw a line,
OA, in some fixed direction, from which we then set
off a series of angles, AOB, AOG, &c. equal to the ob-
* iHkioi, the sun; and /"i-^i'v, to measure.

178 A TREATISE ON ASTRONOMY. [cHAP. V.

served lono;-itiules of the sun tlirouorhont the year, and in
these respective directions measure off from O the dis-
trances OA, OB, OC, &c, representing the distances
deduced from the observed diameter, and then connect
all the extremities A, B, C, &c. of these lines by a con-
tinuous curve, it is evident this will be a correct represen-
tation of the relative orbit of the sun about the earth.
Now, when this is done, a deviation from the circular
figure in the resulting curve becomes apparent ; it is
found to be evidently longer than it is broad — that is to
say, elliptic, and the point O to occupy not the centre,
but one of the foci of the ellipse. The graphical process
here described is sufficient to point out the general figure
of the curve in question ; but for the purposes of exact
verification, it is necessary to recur to the properties of
the ellipse,* and to express the distance of any one of its
points in terms of the angular situation of that point with
respect to the longer axis, or diameter of the ellipse.
This, however, is readUy done ; and when numerically
calculated, on the supposition of the eccentricity being
such as above stated, a perfect coincidence is found to
subsist between the distances thus computed, and those
derived from the measurement of the apparent diameter.
(293.) The mean distance of the eartli and sun being
taken for unity, the extremes are 1'01679 and 0-98321,
But if we compare, in like manner, the mean or average
angidar velocity with the extremes, greatest and least,
we shall find these to be in the proportions of 1 '03386,
1 -00000, and 0-96614, The variation of the sun's aw-
gidar velocity, then, is much greater in proportion than
that of its distance- — fully twice as great ; and if we ex-
amine its numerical expressions at different periods, com-
paring them with the mean value, and also with the cor-
responding distances, it will be found, that, by whatever
fraction of its mean value the distance exceeds the mean,'
the angular velocity will fall short of its mean or average
quantity by very nearly tioice as great a fraction of the
latter, and vice versa. Hence Ave are led to conclude
that the angidar velocity is in the inverse proportion, not
of the distance simply, but of its square : so that, to com^
* Seo Conic Seciions, hv the Rev. H. P. Ilavnilton,

CHAP, v.] LAWS OF ELLIPTIC MOTION. 179

pare the daily motion in longitude of the sun, at one
point, A, of its path, with that at B, we must state the
proportion thus : —

OB^ : OA^ : : daily motion at A : daily motion at B. And
this is found to be exactly verified in every part of the orbit.

(294.) Hence we deduce another remarkable conclu-
sion— viz. that if the sun be supposed really to move
round the circumference of this ellipse, its actual speed
cannot be uniform, but must be greatest at its least dis-
tance, and less at its greatest. For, were it uniform, the
apparent angular velocity Avould be, of course, inversely
proportional to the distance ; simply because the same
linear change of place, being produced in the same time
at different distances from the eye, must, by the laws of
perspective, correspond to apparent angular displacements
inversely as those distances. Since, then, observation
indicates a more rapid law of variation in the angular
velocities, it is evident that mere change of distance, un-
accompanied with a change of actual speed, is insuffi-
cient to account for it ; and that the increased })roximity
of the sun to the earth must be accompanied with an
actual increase of its real velocity of motion along its path.

(295.) This elliptic form of the sun's path, the eccen-
tric position of the earth within it, and the unequal speed
with which it is actually traversed by the sun itself, all
tend to render the calculation of its longitude from theory
(i. e. from a knowledge of the causes and nature of its
motion) difficult, and indeed impossible, so long as the
law oi its actual velocity continues unknown. This laiv,
however, is not immediately apparent. It does not come
forward, as it were, and present itself at once, like the
elliptic form of the orbit, by a direct comparison of an-
gles and distances, but requires an attentive consideration
of the whole series of observations registered during an
entire period. It was not, therefore, without mucli pain-
ful and laborious calculation, that it was discovered by
Kepler (who was also the first to ascertain the elliptic
form of the orbit), and announced in the following terms :
Let a line be always supposed to connect the sun, sup-
posed in motion, with the earth, supposed at rest ; then,
as the sun moves along its ellipse, this line (which is

180 A TREATISE ON ASTRONOMY. [cHAP. V.

called in astronomy the radius vector) will describe or
sweep over that portion of the whole area or surface of
the ellipse which is included between its consecutive
positions : and the motion of the sun will be such that
equal areas are thus swept over by the revolving radius
vector in equal times, in whatever part of the circum-
ference of the ellipse the sim may be moving.

(296.) From this it necessarily follows, that in iin-
equal times, the areas described must be proportional to
the times. Thus, in the figure of art. 292, the time in
which the sun moves from A to B, is the time in which
it moves from C to D, as the area of the elliptic sector
AOB is to the area of the sector DOC.

(297.) The circumstances of the sun's apparent annual
motion may, therefore, be summed up as follows : — It is
performed in an orbit lying in one plane passing through
the earth's centre, called the plane of the ecliptic, and
Avhose projection on the heavens is the great circle so
called. In this plane, however, the actual path is not
circidar, but elliptical ; having the earth, not in its centre,
but in one focus. The eccentricity of this ellipse is 0-01 679,
in parts of a unit equal to the mean distance, or half the
longer diameter of the ellipse ; and the motion of the sun
in its circumference is so regulated, that equal areas of the
ellipse are passed over by the radius vector in equal times.

(298.) What we have here stated supposes no know-
ledge of the sun's actual distance from the earth, nor,
consequently, of the actual dimensions of its orbit, nor
of the body of the sun itself. To come to any conclu-
sions on these points, we must first consider by what
means we can arrive at any knowledge of the distance of
an object to which we have no access. Now, it is ob-
vious, that its parallax alone can afford us any informa-
tion on this subject. Parallax may be generally defined
to be the change of apparent situation of an object
arising from a change of real situation of the observer.
Suppose, then, PABQ to represent the earth, C its centre,
and S the sun, and A, B two situations of a spectator, or,
which comes to the same thing, the stations of two spec-
tators, both observing the sun S at the same instant. The
spectator A will see it in the direction ASa, and will re-

fchAt". V.J DIURNAL oft GEOCENTRIC PARALLAX. 181

fer it to a point a in the infinitely distant sphere of the
fixed stars, while the spectator B will see it in the direc-^

/

tion BSi, and refer it to h. The angle included between
these directions, or the measure of the celestial arc a b, by
which it is displaced, is equal to the angle ASB ; and if
this angle be known, and the local situations of A and B,
with the part of the earth's surface AB included between
them, it is evident that the distance CS may be calculated.

(299.) Parallax, however, in the astronomical accepta-
tion of tlie word, has a more technical meaning. It is
restricted to the difference of apparent positions of any
celestial object when viewed from a station on the stir-
face of the earth, and from its centre. The centre of
the earth is the general station to which all astronomical
observations are referred : but, as we observe from the
surface, a reduction to the centre is needed ; and the
amount of this reduction is called parallax. Thus, thd
sun being seen from the earth's centre, in the direction
CS, and from A on the surface in the direction AS, the
angle ASC, included between these two directions, is the
parallax at A, and similarly BSC is that at B.

Parallax, in this sense, may be distinguished by the
epithet diurnal, or geocentric, to discriminate it frolii
the annical, or heliocentric ; of which more hereafteri

(300.) The reduction for parallax, then, in any pro-
posed case, is obtained from the consideration of the
triangle ACS, formed by the spectator, the centre of the
earth, and the object observed; and since the side CA
prolonged passes through the observer's zenith, it ia
evident that the effect of parallax, in this its technical
acceptation, is always to depress the object observed ill
a vertical circle. To estimate the amount of this de«
pression, we have, by plane trigonometry,
CS : CA : : sine of CAS=sine of ZAS : sine of ASC.

Q w

182 A TREATISE ON ASTRONOMY. [cHAP. V,

(301.) The parallax, then, for objects equidistant from
the earth, is proportional to the sines of their zenith dis-
tances. It is, therefore, at its maximum when the body
observed is in the horizon. In this situation it is called
the horizonful parallax ; and when this is known, since
small arcs are proportional to their sines, the parallax at
any given altitude is easily had by the following rule : —

Parallax = (horizontal parallax) x sine of zenith dis-
tance;

The horizontal parallax is given by this proportion : —

Distance of object : earth's radius : : rad. : sine of ho-
rizontal parallax.

It is, therefore, known, when the proportion of the
object's distance to the radius of the earth is known;
and vice versa — if by any method of observation we can
come at a knowledge of the horizontal parallax of an
object, its distance, expressed in units equal to the earth's
radius, becomes known.

(302.) To apply this general reasoning to the case of
the Sim. Suppose two observers — one in the northern,
the other in the southern hemisphere — at stations on the
same meridian, to observe on the same day the meridian
altitudes of the sun's centre. Having thence derived
the apparent zenith distances, and cleared them of the
effects of refraction, if the distance of the sun w^ere equal
to that of the fixed stars, the sum of the zenith distances
thus found would be precisely equal to tlie sum of the
latitudes north and south of the places of observation.
For the sum in question would then be equal to tlie
angle %C]ji, which is the meridional distance of the
stations across the equator. Bat the effect of parallax
being in both cases to increase the apparent zenith dis-
tances, their observed sum will be greater than the sum
of the latitudes, by the whole amount of the two paral-
laxes, or by the angle ASB. This angle, then, is
obtained by subducting the sum of the latitudes from
that of the zenith distances ; and this once determined,
the horizontal parallax is easily found, by dividing the
angle so determined by the sum of the sines of the two
..lafitu^es. :c '"^ - f / ,. fj i j ^ ..^ , ^ ,- ;

(3Q3.) If the two stations be not exactly on the same

CHAP. V.J PARALLAX OF THE SUN. 183

meridian (a condition A'ery difficult to fulfil), the same
process will apply, if we take care to allow for the
change of the sun's actual zenith distance in the interval
of time elapsing between its arrival on the meridians of
the stations. This change is readily ascertained, either
from tables of the sun's motion, grounded on the ex-
perience of a long course of observations, or by actual
observation of its meridional altitude on several days
before and after that on Avhich the observations for paral-
lax are taken. Of course, the nearer the stations are to
each other in longitude, the less is this interval of time ;
and, consequently, the smaller the amount of this correc-
tion ; and, therefore, the less injurious to the accuracy
of the final result is any uncertainty in the daily change
of zenith distance which may arise from imperfection
in the solar tables, or in the observations made to deter-
mine it.

(304.) The horizontal parallax of the sun has been
concluded from observations of the nature above de-
scribed, performed in stations the most remote from each
otlier in latitude, at which observatories have been in-
stituted. It has also been deduced from other methods
of a more refined nature, and susceptible of much greater
exactness, to be hereafter described. Its amount, so
obtained, is about 8"*6. Minute as this quantity is,
there can be no doubt that it is a tolerably correct ap-
proximation to the truth ; and in conformity with it, we
must admit the sun to be situated at a mean distance from
us, of no less than 23,984 times the length of the earth's
radius, or about 95,000,000 miles.

(305.) That at so vast a distance the sun should ap-
■ pear to us of the size it does, and should so powerfully
influence our condition by its heat and light, requires us
to form a very grand conception of its actual magnitude,
and of the scale on which those important processes are
carried on within it, by which it is enabled to keep up its
liberal and unceasing supply of these elements. As to
its actual magnitude we can be at no loss, knowing its
distance, and the angles under which its diameter appears
to us. An object, placed at the distance of 95,000,000
miles, and subtending an angle of 32' 3", must have a

184 A TREATISE ON ASTRONOMY. [cHAP. V.

real diameter of 882,000 miles. Such, then, is the dia-.
meter of this stupendous glolie. If we compare it with
what we have already ascertained of the dimensions of
our own, Ave shall find that in linear magnitude it exceeds
the earth in the proportion of III2 to 1, and in bulk in
that of 1,-384,472 to 1.

(306.) It is hardly possible to avoid associating our
ooncepticn of an object of definite globular figure, and of
such enormous dimensions, with some corresponding
?tttribute of massiveness and material solidity. That the
gun is not a mere phantom, but a body having its own
peculiar structure a«d economy, our telescopes distinctly
tnform us. They show us dark spots on its surface,
vyhich slowly change their places and forms, and by
attending to whose situation, at different times, astrono^
triers have ascertained that the sun revolves about an
axis inclined at a constant angle of 82° 40' to tlie plane
of the ecliptic, performing one rotation in a period of 25
days and in the same direction with the diurnal rotation
af tlie earth, i. e. from west to east. Here, then, we
liave an analogy with our own globe ; the slower and
more majestic movement only corresponding with the
greater dimensions of the machinery, and impressing us
with the prevalence of similar mechanical laws, and of,
at least, such a community of nature as the existence of
inei-tia and obedience to force may argue. Now, in the
exact proportion in which we invest our idea of this im-i
mense bulk with the attribute of inei-tia, or Aveight, it be-
comes difficult to conceive its circulation round so com^
paratively small a body as the earth, without, on the one
hand, dragging it along, and displacing it, if bound to it
by some invisible tie ; or, on the other hand, if not so
held to it, pursuing its course alone in space, and leaving-
^he earth behind. If v/e tie two stones together by a
string, and fling them aloft, we see them circulate about
a point between them, which is their common centre of
gravity ; but if one of them be gi'eatly more ponderous
than the other, this common centre will be proportionally
nearer to that one, and even within its surface, so that the
smaller one will circulate, in fort, about the larger, which
will be comparatively but little disturbed from its place.

DIMENSIONS AND nOTATION OF THE EARTH. 185

(307.) Whether the earth move round the sun, the sun
round the earth, or both round their common centre of
gravity, will make no difference, so far as appearances are
concerned provided the stars be supposed sufficiently dis-
tant to undergo no sensilde apparent parallactic displace-
ment by the motion so attributed to the earth. Whether
they are so or not must still be a matter of inquiry ; and
from the absence of any measureable amount of such dis-
placement, we can conclude nothing but this, that the
scale of the sidereal universe is so great, that the mutual
orbit of the earth and sun may be regarded as an imper-
ceptible point in its comparison. Admitting, then, in
conformity with the laws of dynamics, that two bodies
connected with and revolving al^out each other in free
space do, in fact, revolve about their common centre of
gravity, Avhich remains immoveable by their mutual ac-
tion, it becomes a matter of further inquiry, U'hereahouls
between them the centre is situated. Mechanics teaches
us that its place will divide their mutual distance in the
inverse ratio of their iveights or masses ;* and calculations
grounded on phenomena, of which an account will be
given further on, inform us that this ratio, in the case of
the sun and earth, is actually that of 354,936 to 1, — the
sun being, in that proportion, more ponderous than the
earth. From this it will follow tliat the common point
about which they both circulate is only 267 miles from the
sun's centre, or about ^^^^ g^th part of its own diameter.

(308.) Henceforward, then, in conformity with the
above statements, and with the Copernican view of our
system, we must learn to look upon the sun as the com-
paratively motionless centre about which the earth per-
forms an annual elliptic orbit of the dimensions and ec-
centricity, and with a velocity regulated according to the
law above assigned ; the sun occupying one of the foci
of the ellipse, and from that station quietly disseminating
on all sides its light and heat ; while the earth, travelling
round it, and presenting itself differently to it at different
times of the year and day, passes through the varieties of
day and night, summer and winter, which we enjoy.
(309.) In this annual motion of the earth, its axis pre-
* See Cab. Cyc. Mechanics, Centre of Gravity.

a2

186 A TREATISE ON ASTRONOMY. [cHAP. V.

serves, fit all times, the same direction as if the orbitiial
movement liad no existence ; and is carrieel round paral-
lel to itself, and pointing always to the same vanishing
point in the sphere of the fixed stars. This it is which
gives rise to the variety of seasons, as we shall now ex-
plain. In so doing, we shall neglect (for a reason which
"will be presently explained) tlie ellipticity of the orbit,
and suppose it a circle, with the sun in the centre.

(310.) Let, then; S represent the sun, and A, B, C, D,
four positions of the earth in its orbit, 90° apart, viz. A

• that which it has on the 21st of March, or at the time of
the vernal equinox ; B that of the 21st of June, or the
summer solstice ; C that of the 21st of September, or the
autumnal equinox ; and D that of the 21st of December,
or the winter solstice. In each of these positions let PQ
represent the axis of the earth about Avhich its diurnal
Yotation is performed without interfering with its annual
motion in its orbit. Then, since the sun can only en^
lighten one half of the surface at once, viz. that turned
towards it, the shaded portions of the globe in its several
positions will represent the dark, and the bright, the en--
lightened halves of the earth's surface in these positions.
Now, 1st, in the position A, the sun is vertically over tlie
intersection of the equinoctial FE and the ecliptic IIG,
It is, therefore, in the equinox ; and in this position the
poles P, Q, both fall on the extreme confines of the en-,
lightened side. In this position, therefore, it is day over
half the northern and half the southern hemisphere at
once ; and as the earth revolves on its axis, every point
of its surface describes half its diunial course in light, ajicl

CHAP, v.] TEMPERATURE OF THE EARTH. 187

half in darkness ; in other words, the duration of day and
night is here equal over the whole globe : hence the term
equinox. The same holds good at the autumnal equinox
on the position C.

(311.) B is the position of the earth at the time of the
northern, summer solstice. Here the north pole P, and
a considerable portion of the earth's surface in its neigh-
bourhood, as far as B, are situated within the enlighten-
ed half. As the earth turns on its axis in this position,
therefore, the whole of that part remains constantly en-
lightened ; therefore, at this point of its orbit, or at this
season of the year, it is continual day at the north pole,
and in all that region of the earth which encircles this
pole as far as B — that, is, to the distance of 23° 28' from
the pole, or within what is called, in geography, the arctic
circle. On the other hand, the opposite or south pole Q,
with all the region comprised within the antarctic circle,
as far as 23° 28' from the south pole, are immersed at
this season in darkness, during the entire diurnal rotation,
so that it is here continual night.

(312.) With regard to that portion of the surface com-
prehended between the arctic and antarctic circles, it is
no less evident that the nearer any point is to the north
pole, the larger will be the portion of its diurnal course
comprised within the bright, and the smaller within the
dark hemisphere ; that is to say, the longer will be its
day, and the shorter its night. Every station north of the
equator will have a day of more and a night of less than
twelve hour's duration, and vice versa. All these phe-
nomena are exactly inverted when the earth comes to the
opposite point D of its orbit.

(313.) Now, the temperature of any part of the earth's
surface depends mainly, if not entirely, on its exposure to
the sun's rays. Whenever the sun is above the horizon
of any place, that place is receiving heat ; when below,
parting with it, by the process called radiation ; and the
whole quantities received and parted with in the year
must balance each othfer at every station, or the equilibri-
um of temperature would not be supported. Whenever,
then, the sun remains more than twelve hours above the
horizon of any place, and less beneath, the general tempe-

188 A TREATISE ON A6TR0N0MY. [cHAP. V.

rature of that place will be above the average ; when the
reverse, below. As the eartli, tlien, moves from A to B,
the days growing longer, and the nights shorter in the
northern hemisphere, the temperature of every part of that
hemisphere increases, and we pass from spring to sum-
mer, while at the same time the reverse obtains in the
southei'n hemisphere. As the earth passes from B to C,
the days and nights again approach to equality — the ex-
cess of temperature in the northern hemisphere above the
mean state grows less, as well as its defect in the south-
ern ; and at the autumnal equinox, C, the mean state is
once more attained. From thence to D, and, finally,
round again to A, all the same phenomena, it is obvious,
must again occur, but reversed, it being now Avinter in
the nortliern, and summer in the southern hemisphere.

(314.) All this is exactly consonant to observed fact.
The continual day within the polar circles in summer,
and night in winter, the general increase of temperature
and length of day as the sun approaches the elevated
pole, and the reversal of the seasons in the northern and
southern hemispheres, are all facts too well known to
require further comment. The positions A, C of the
earth correspond, as we have said, to the equinoxes ;
those at B, D to the solstices. This term must be ex-
plained. If, at any point, X, of the orbit, we draw XP
the earth's axis, and XS to the sun, it is evident that the
angle PXS will be the sun's ^joZ«j- distance. Now, this
angle is at its maximum in the position D, and at its
minimum at B; being in the former case =90° + 23''
28' = 103° 28', and in the latter 90°— 23^ 28' = 66° 33'.
At these points the sun ceases to approach to or to recede
from the pole, and hence the name solstice.

(315.) The elliptic form of the earth's orbit has but
a very trifling share in producing the variation of tem-
perature corresponding to the difference of seasons. This
assertion may at first sight seem incompatible with what
we know of the laws of the communication of heat from
a luminary placed at a variable "distance. Heat, like
light, being equally dispersed from the sun in all direc-
tions, and being spread over the surface of a sphere con-
tinually enlarging as we recede from the centre, must of

CHAP, v.] EQUAL DISTRIBUTION OF HEAT. • 189

course diminish in intensity according to the inverse pro-
portion of the surface of the sphere over which it is
spread ; that is, in the inverse proportion of the square
of the distance. But Ave have seen (art. 293) that this
is also the proportion in which the angular velocity of
the earth about the sun varies. Hence it appears, that
tlie momentary supply of heat received by the earth from
the sun varies in the exact proportion of the angular ve-
locity, i. e. oi the momentary increase of lotigitude ; and
from this it follows, that equal amounts of heat are re-
ceived from the sun in passing over equal angles round
it, in whatever part of the ellipse those angles may be
gituated. Let, then, S represent the sun ; AQMP the

earth's orbit ; A its nearest point to the sun, or, as it is
called, the perihelion of its orbit ; M the farthest, or tho
aphelion; and therefore ASM the axis of the ellipse.
Now, suppose the orbit divided into two segments by a
straight line PSQ drawn through the sun, and any how
situated as to direction ; then, if we suppose the earth
to circulate in the direction PAQiMP, it will have passed
over 180° of longitude in moving from P to Q, and as
many in moving from Q to P. It appears, therefore,
from what has been shown, that the supplies of heat re^
ceived from the sun will be equal in the two segments,
in whatever direction the line PSQ be drawn. They
will, indeed, be described in unequal times ; that in
which the perihelion A lies in a shorter, and the other
in a longer, in proportion to their unequal area ; but the
greater proximity of the sun in the smaller segment cow^

190 A TREATISE ON ASTRONOMY. [cHAP. V.

pensates exactly for its more rapid description, and thus
an equilibrium of heat is, at it were, maintained. Were
it not for this, the eccentricity of the orbit would mate-
rially influence the transition of seasons. The fluctua-
tion of distance amounts to nearly -joth of its mean quan-
tity, and consequently, the fluctuation in the sun's direct
heating power to double this, or ^jih of the whole.
Now, the perihelion of the orbit is situated nearly at the
place of the northern winter solstice ; so that, were it
not for the compensation we have just described, the
effect would be to exaggerate the difl'erence of summer
and winter in the southern hemisphere, and to moderate
it in the northern ; thus producing a more violent alter-
nation of climate in the one hemisphere, and an approach
to perpetual spring in the other. As it is, however, no
such inequality subsists, but an equal and impartial dis-
tribution of heat and light is accorded to both.*

(316.) The great key to simplicity of conception in
astronomy, and, indeed, in all sciences where motion is
concerned, consists in contemplating every movement as
referred to points which are either permanently fixed,
or so nearly so, as that their motions shall be too small
to interfere materially with and confuse our notions. In
the choice of these primary points of reference, too, we
must endeavour, as far as possible, to select such as have
simple and symmetrical geometrical relations of situa-
tion with respect to the curves described by the moving
parts of the system, and which are thereby fitted to per-
form the oflfice of natural centr«3S — advantageous sta-
tions for the eye of reason and theory. Having learned
to attribute an orbilual motion to the earth, it loses this
advantage, which is transferred to the sun, as the fixed
centre about which its orbit is performed. Precisely as,
when embarrassed by the earth's diurnal motion, we
have learned to transfer, in imagination, our station of
observation from its surface to its centre, by the appli-
cation of the diurnal parallax ; so, when we come to in-
quire into the movements of the planets, Ave shall find

* See Geological Transactions, 1832, " On the Astronomical Causes
which may iiilluence Geological Phenomena," by the author of thia
work.

MEAN AND TRUE LONGITUDE OF THE SUN. 191

ourselves continually embarrassed by the orbitual mo-
tion of our point of view, unless, by the consideration of
the annual or heliocentric para/lax, as it may be termed,
we consent to refer all our observations on them to the
centre of the sun, or rather to the common centime of gi'a-
vity of the sun, and the other bodies which are connect-
ed with it in our system. Hence arises the distinction
between the geocentric and heliocentric place of an ob-
ject. The former refers its situation in space to an
imaginary sphere of infinite radius, having the centre of
the earth for its centre — the latter to one concentric with
the sun. Thus, when we speak of the heliocentric lon-
gitudes and latitudes of objects, we suppose the specta-
tor situated in the sun, and referring them, by circles
perpendicular to the plane of the ecliptic, to the great
circle marked out in the heavens by the infinite prolonga-
tion of that plane.

(317.) The point in the imaginary concave of an in-
finite heaven, to which a spectator in the sun refers the
earth, must, of course, be diametrically opposite to that to
which a spectator on the earth refers tlie sun's centre ;
consequently, the heliocentric latitude of the earth is
always nothing, and its heliocentric longitude always
equal to the sun's geocentric longitude +180°. The
heliocentric equinoxes and solstices are, therefore, the
same as the geocentric ; and to conceive them, we have
only to imagine a plane passing through the sun's centre,
parallel to the earth's equator, and prolonged infinitely
on all sides. The line of intersection of this plane and
the plane of the ecliptic is the line of equinoxes, and the
solstices are 90° distant from it.

(318.) The position of the longer axis of the earth's
orbit is a point of great importance. In the figure (art.
315) let ECLI be the ecliptic, E the vernal equinox, L
the autumnal {i. e. the points to which the earth is re-
ferred from the sun ivhen its heliocentric longitudes are
0° and 180° respectively). Supposing the earth's mo-
tion to be performed in the direction ECLI, the angle
ESA, or the longitude of the perihelion, in the year 1800
was 99° 30' 5" : Ave say in the year 1800, because, in
point of fact, by the operation of causes hereafter to be

1^2 A TREATISE ON ASTRONOMY. [cHAP. V.

explained, its position is snbject to an extremely slow va-
riation of about 12" per annum to the eastward, and
which, in the progress of an immensely long period — of no
less than 20,984 years — carries the axis ASM of the
orbit completely round the whole circumference of the
ecliptic. But this motion must be disregarded for the
present, as well as many other minute deviations, to be
brought into view when they can be better understood.
(319.) Were the earth's orbit a circle, described with
a uniform velocity about the sun placed in its centre, no-
thing could be easier than to calculate its position at any
time, with respect to the line of equinoxes, or its longi-
tude, for we should only have to reduce to numbers the
proportion following; viz. One year :the time elapsed ::
360° : the arc of longitude passed over. The longitude
so calculated is called in astronomy the vnerm longitude of
the earth. But since the earth's orbit is neither circular,
nor uniformly described, this rule will not give us the true
place in the orbit at any proposed moment. Neverthe-
less, as the eccentricity and deviation from the circle are
small, the true place will never deviate very far from that
so determined (which, for distinction's sake, is called the
mean place), and the former may at all times be calculated
from the latter, by applying to it a correction or equation
(as it is termed), whose amount is never very great, and
whose computation is a question of pure geometry, de-
pending on the equable description of areas by the earth
about the sun. For since, in the elliptic motion, accord-
ing to Kepler's law above stated, areas not angles are
described uniformly, the proportion must now be stated
thus ; One year : the time elapsed : : the whole area of
the ellipse : the area of the sector swept over by the ra-
dius vector in that time. This area, therefore, becomes
known, and it is then; as above observed, a problem of
pure geometry to ascertain the angle about the sun (ASP,
fig. art. 315), which corresponds to any proposed frac-
tional area of the whole ellipse supposed to be contained
in the sector APS. Suppose we set out from A the pe-
rihelion, then will the angle ASP at first increase more
rapidly than the mean longitude, and will, therefore, du"
ring the whole semi-revolution from A to M, exceed it ia

Mean and True longitude of the sun. 193

amount ; or, in other words, the trite place will be in ad-
vance of the mean : at M, one half of the year will have
elapsed, and one half the orbit have been described,
whether it be circular or elliptic. Here, then, the mean
and true places coincide ; but in all the other half of the
orbit, from M to A, the true place will fall short of the
mean, since at M the angular motion is slowest, and the
true place from this point begins to lag behind the mean
• — to make up with it, however, as it approaches A, where
it once more overtakes it.

(320.) The quantity by which the true longitude of the
earth differs from the mean longitude is called the equa-
tion of the centre, and is additive during all the half-year
in which the earth passes from A to M, beginning at 0°
0' 0", increasing to a maximum, and again diminish-
ing to zero at M ; after which it becomes subtractive,
attains a maximum of subtractive magnitude between M
and A, and again diminishes to 0 at A. Its maximum,
both additive and subtractive, is 1° 55' 33"-3.

(321.) By applying, then, to the earth's mean longi-
tude, the equation of the centre corresponding to any
given time at which we would ascertain its place, the true
longitude becomes known ; and since the sun is always
seen from the earth in 180° more longitude than the earth
from the sun, in this way also the sun's true place in the
ecliptic becomes known. The calculation of the equa-
tion of the centre is performed by a table constructed for
that purpose, to be found in all " Solar Tables."

(322.) The maximum value of the equation of the cen-
tre depends only on the ellipticity of the orbit, and may
be expressed in terms of the eccentricity. J^ice versa,
therefore, if the former quantity can be ascertained by
observation, the latter may be derived from it ; because,
whenever the law, or numerical connexion, between two
quantities is known, the one can always be determined
from the other. Now, by assiduous observation of the
sun's transits over the meridian, we can ascertain, for
every day, its exact right ascension, and thence conclude
its longitude (art. 260). After this, it is easy to assign the
angle by which this observed longitude exceeds or falls
short of the mean ; and the greatest amount of this excess

R

194 A TREATISE ON ASTRONOMY. l^CHAP. V.

or defect which occurs in the whole year, is the maxi-
mum equation of the centre. This, as a means of ascer-
taining the eccentricity of the orbit, is a far more easy and
accurate method than that of conch;ding its distance by
measuring its apparent diameter. Tlie results of the two
methods coincide, however, perfectly.

(323.) If the ecliptic coincided Avith the equinoctial,
the effect of the equation of the centre, by disturbing the
uniformity of the sun' s apparent motion in longitude,
would cause an inequality in its time of coming on the
meridian on successive days. When the sun's centre
comes to the meridian, it is apparent yjoon, and if its mo-
tion in longitude were uniform, and the ecliptic coincident
with the equinoctial, this would always coincide with
mean noon, or the stroke of 12 on a well-regulated solar
clock. But, independent of the want of uniformity in
its motion, the obliquity of the ecliptic gives rise to an-
other inequality in tliis respect ; in consequence of which
the sun, even supposing its motion in the ecliptic uniform,
would yet alternately, in its time of attaining the meri-
dian, anticipate and fall short of the mean noon as shown
by the clock. For the right ascension of a celestial ob-
ject, forming a side of a right-angled spherical trian-
gle, of which its longitude is the hypothenuse, it is
clear that the uniform increase of the latter must necessi-
tate a deviation from uniformity in the increase of the
former.

(324.) These two causes, then, acting conjointly, pro-
duce, in fact, a very considerable fluctuation in the time
as shown per clock, v/hen the sun really attains the
meridian. It amounts, in fact, to upwards of half an
hour ; apparent noon sometimes taking place as much as
16} min. before mean noon, and at others as much as 14i
min. after. This difference between apparent and mean
noon is called the equation of time, and is calculated and
inserted in ephemerides for every day of the year, under
that title ; or else, which comes to the same thing,
the moment, in mean time, of the sun's culmination,
for each day, is set down as an astronomical phenome-
non to be observed.

(325.) As the sun, in its apparent annual course, is

CHAP, v.] TROPICAL AND ANOMALISTIC YEARS. 195

carried along the ecliptic, its declination is continually
varying- between the extreme limits of 23° 2^' 40" north,
and as much south, which it attains at the solstices. It
is consequently always vertical over some part or other
of that zone or belt of the earth's surface which lies be-
tween the north and south parallels of 23° 28' 40".
These parallels are called in geography the tropics ; the
northern one that of Cancer, and the southern of Capri-
corn; because the sun, at the respective solstices, is situ-
ated in the division or signs of the ecliptic so denomi-
nated. Of these signs there are twelve, each occupying
30° of its circumference. They commence at the vernal
equinox, and are named in order — Aries, Taurus, Gemi-
ni, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Ca-
pricornus, Aquarius, Pisces. They are denoted also by
the following symbols:— T, 8, H, 25, ^, n, ^ ,"1, /,
\3, ^, >£. The ecliptic itself is also divided into
signs, degrees, and minutes, &c. thus, 5' 27° 0' corres-
ponds to 177° 0' ; but this is beginning to be disused.

(326.) When the sun is in either tropic, it enlightens,
as we have seen, the pole on that side the equator, and
shines over or beyond it to the extent of 23° 2!^' 40".
The parallels of latitude, at this distance from either
pole, are called the polar circles, and are distinguished
from each other by the names arctic and antarctic. The
regions within these circles are sometimes termed frigid
zones, while the belt between the tropics is called the
torrid zone, and the immediate belts temperate zones.
These last, however, are merely names given for the
sake of naming ; as, in fact, owing to the different dis-
tribution of land and sea in the two hemispheres,
zones of climate are not co-terminal with zones of lati-
tude.

(327.) Our seasons are determined by the apparent
passages of the sun across the equinoctial, and its alter-
nate arrival in the northern and southern hemisphere.
Were the equinox invariable, this Avould happen at in-
tervals precisely equal to the duration of the sidereal
year ; but, in fact, owing to the slow conical motion of
the earth's axis described in art. 2G4, the equinox re-
treats on the ecliptic, and meets the advancing sun somC'

196 A TREATISE ON ASTRONOMY. [cHAP. V,

what before the whole sidereal circuit is completed. The
annual retreat of the equinox is 50"-l, and this arc is
described by the sun in the ecliptic in 20' 19"-9. By
so much shorter, then, is the periodical return of our
seasons than the true sidereal revolution of the earth
round the sun. As the latter period, or sidereal year, is
equal to SeS"* 6*' 9™ 9' -6, it follows, then, that the former
must be only 365' 5'' 48'" 49' -7 ; and this is what is meant
by the tropical year.

(328.) We have already mentioned that the longer
axis of the ellipse described by the earth lias a slow mo-
tion of 11 "'8 per annum in advance. From this it re-
sults, that when the earth, setting out from the perihelion,
has completed one sidereal period, the perihelion will
have moved forward by 11""8, which arc must be de-
scribed before it can again reach the perihelion. In so
doing, it occupies 4' 39"*7, and this must therefore be
added to the sidereal period, to give the interval between
two consecutive returns to the perihelion. This in-
terval, then, is 365'' 6*^ IS"" 49' -3,* and is what is called
the ano7nalistic year. All these periods have their uses
in astronomy ; but that in which mankind in general are
most interested is the tropical year, on which the return
of tlie seasons depends, and which we thus perceive to
be a compound phenomenon, depending chiefly and di-
rectly on the annual revolution of the earth round the
sun, but subordinately also, and indirectly, on its rota-
tion round its own axis, which is what occasions the
precession of the equinoxes ; thus aflfording an instruc-
tive example of the way in which a motion, once ad-
mitted in any part of our system, may be traced in its
influence on others with which at first sight it could not
possibly be supposed to have any thing to do.

(329.) As a rough consideration of the appearance of
the earth points out the general roundness of its form,
and more exact inquiry has led us first to the discovery
of its elliptic figure, and, in the further progress of re-
finement, to the perception of minuter local deviations

* These numbers, as well as all the other numerical data of our sys-
tem, are taken from Mr. Baily's Astronomical Tables and FormuLse un»
less the contrary is expressed-

CHAP, v.] PHYSICAL COXSTITUTION OF THE SUN. 197

from that figure ; so, in investigating the solar motions,
the first notion we obtain is that of an orbit, generally
speaking, round, and not far from a circle, which, on
more careful and exact examination, proves to be an
ellipse of small eccentricity, and described in conformity
with certain laws, as above stated. Still minuter in-
(5[uiry, however, detects yet smaller deviations again
from this form and from these laws, of which we have
a specimen in the slow motion of the axis of the orbit
spoken of in art. 318 ; and which are generally compre-
hended under the name of perturbations and secular in-
equalities. Of these deviations, and their causes, we
shall speak hereafter at length. It is the triumph of
physical astronomy to have rendered a complete account
of them all, and to have left nothing unexplained, either
in the motions of the sun or in those of any other of the
bodies of our system. But the nature of this explana-
tion cannot be understood till we have developed the
law of gravitation, and carried it into its more direct
consequences. This will be the object of our three fol-
lowing chapters ; in v>diich we shall take advantage of
the proximity of the moon, and its immediate connexion
with and dependence on the earth, to render it, as it
were, a stepping-stone to the general explanation of the
planetary movements.

(330.) We shall conclude this by describing what is
known of the physical constitution of the sun.

When viewed through powerful telescopes, provided
with coloured glasses, to take off the heat, which would
otherwise injure our eyes, it is observed to have fre-
quently large and perfectly black spots upon it, sur-
rounded Avith a kind of border, less completely dark,
called a penumbra. Some of these are represented at
a, b, c, plate iii. fig. 1, in the plate at the end of this
volume. They are, however, not permanent. When
watched from day to day, or even from hour to hour,
they appear to enlarge or contract, to change their forms,
and at length to disappear altogether, or to break out
anew in parts of the surface where none were before.
In such cases of disappearance, the central dark spot
always contracts into a point, and vanishes before the

r2

198 A TREATISE ON ASTRONOMY. [cHAP. V.

border. Occasionally they break up, or divide into two
or more, and in those offer every evidence of that ex-
treme mobility which belongs only to the fluid state, and
of that excessively violent agitation which seems only
compatible with the atmospheric or gaseous state of mat-
ter. The scale on which their movements take place is
immense. A single second of angular measure, as seen
from the earth, corresponds on the sun's disc to 465
miles ; and a circle of this diameter (containing there-
fore nearly 220,000 square miles) is the least space which
can be distinctly discerned on the sun as a visible area.
Spots have been observed, however, whose linear dia-
meter has been upwards of 45,000 miles ;* and even, if
some records are to be trusted, of very much greater ex-
tent. That such a spot should close up in six weeks'
time (for they hardly ever last longer), its borders must
approach at the rate of more than 1000 miles a day.

Many other circumstances tend to corroborate this
view of the subject. The part of the sun's disc not oc-
cupied by spots is far from uniformly bright. Its ground
is finely mottled with an appearance of minute, dark
dots, or pores, which, when attentively watched, are
found to be in a constant state of change. There is
nothing which represents so faithfully this appearance
as the slow subsidence of some flocculent chymical pre-
cipitates in a transparent fluid, Avhen viewed perpen-
dicularly from above : so faithfully, indeed, that it is
hardly possible not to be impressed with the idea of a
luminous medium intermixed, bvit not confounded, with
a transparent and non-luminous atmosphere, either float-
ing as clouds in our air, or pervading it in vast sheets
and columns like flame, or the streamers of our northern
lights.

(331.) Lastly, in the neighbourhood of great spots, or
extensive groups of them, large spaces of the surface are
often observed to be covered with strongly marked
curved, or branching streaks, more luminous than the
rest, called facidse, and among these, if not already
existing, spots frequently break out. They may,

* Mayer, Obs. Mar. 15, 1758. " Ingens macula in sole conspiciebatup
ciyus diameter = ^V ^^^^' soUs."

CHAP. V«J PROBABLE NATURE OF THE SOLAR SPOTS. 199

perhaps, be regarded with most probability, as the
ridges of immense waves in the hnninous regions of the
sun's atmosphere, indicative of violent agitation in their
neighbourhood.

(332.) But what are the spots ? Many fanciful notions
have been broached on this subject, but only one seems
to have any degree of physical probability, viz. that they
are the dark, or at least comparatively dark, solid body
of the sun itself, laid bare to our view by those immense
fluctuations in the luminous regions of its atmosphere, to
which it appears to be subject. Respecting tlie manner
in which this disclosure takes place, different ideas again
have been advocated. Lalande (art. 3240) suggests,
that eminences in the nature of mountains are actually
laid bare, and project above the luminous ocean, appear-
ing black above it, while their shoaling declivities pro-
duce the penumbrae, where the luminous fluid is less
deep. A fatal objection to this theory is the perfectly
uniform shade of the penumbra and its sharp termination,
both inwards, where it joins the spot, and outwards,
where it borders on the bright surface. A more proba-
ble view has been taken by Sir William Herschel,* who
considers the luminous strata of the atmosphere to be
sustained far above the level of the solid body by a
transparent elastic medium, carrying on its upper sur-
face {or rather, to avoid the former objection, at some
considerably loiver level within its depth,) a cloudy
stratum which, being strongly illuminated from above,
reflects a considerable portion of the light to our eyes,
and forms a penumbra, while the solid body, shaded by
the clouds, reflects none. The temporary removal of
both the strata, but more of the upper than the lower, he
supposes effected by powerful upward currents of the
atmosphere, arising, perhaps, from spiracles in the body,
or from local agitations. See fig. 1. d, plate III.

(333.) The region of the spots is confined within
about 30° of the sun's equator, and, from their motion on
the surface, carefully measured with micrometers, is as-
certained the position of the equator, which is a plane
inclined 7° 30' to the ecliptic, and intersecting it in a line
*Phil. Trans. 1801.

200 A TREATISE ON ASTRONOMY. [cHAP. V.

whose direction makes an angle of 80^ 21' with that ol
the equinoxes. It has been also noticed (not, we think,
without great need of further confirmation), that extinct
spots have again broken out, after long intervals of time,
on the same identical points of the sun's globe. Our
knowledge of the period of its rotation (which, according
to Delambre's calculations, is 25''*01154, but, according
to others, materially different,) can hardly be regarded as
sufficiently precise to establish a point of so much nicety.
(334.) That the temperature at the visible surface of
the sun cannot be otherwise than very elevated, much
more so than any artificial heat produced in our furnaces,
or by chemical or galvanic processes, we have indications
of several distinct kinds : 1st, From the law of decrease
of radiant heat and light, which, being inversely as the
squares of the distances, it follows, that the heat received
on a given area exposed at the distance of the earth, and
on an equal area at the visible surface of the sun, must
be in the proportion of the area of the sky occupied by
the sun's apparent disc to the whole hemisphere, or as 1
to about 300000. A far less intensity of solar radiation,
collected in the focus of a burning glass, suffices to dis-
sipate gold and platina in vapour. 2dly, From the fa-
cility with whicli the calorific rays of the sun traverse
glass, a property which is found to belong to the heat of
artificial fires in the direct proportion of their intensity.*
3dly, From the fact, that the most vivid flames disappear,
and the most intensely ignited solids appear only as black
spots on the disk of the sun when held between it and
the eye.f From this last remark it follows, that the body
of the sun, hov/ever dark it may appear when seen through
its spots, 7nay, nevertheless, be in a state of most intense

* By direct measurement with the ac/Zrtome/er, an instrument I have
long employed in such inquiries, and whose indications are hable to none
of those sources of fallacy whicli beset the usual modes of estimation, I
find that out of 1000 calorific solar rays, 816 penetrate a sheet of plate
glass 012 inch thick; and that of 1000 rays which have passed through
one such plate, 859 are capable of passing through another.-— A «;/ior.

t The ball of ignited quick-lirae, in Lieutenant Drummond's oxy-hydro-
gen lamp, gives the nearest imitation of the solar splendour which has
yet been produced. The appearance of this against the sun was, how-
ever, as described in an imperfect trial at which I was present. The
experiment ought to be repeated under favourable circumstances.—
Autlior

CHAP, v.] ACTION OF THE SUN's RAYS ON THE EARTH. 201

ignition. It does not, however, follow of necessity that
it must be so. The contrary is at least physically possi-
ble. A perfectly reflective canopy would effectually de-
fend it from the radiation of the luminous regions above
its atmosphere, and no heat would be conducted down-
wards through a gaseous medium increasing rapidly in
density. That the penumbral clouds are highly reflect-
ive, the fact of their visibility in such a situation can
leave no doul)t.

(33.5.) This immense escape of heat by radiation, we
may also remark, will fully explain the constant state of
tumultuous agitation in Avhich tlie fluids composing the
visible surface are maintained, and the continual genera-
tion and filling in of the pores, without having recourse
to internal causes. The mode of action here alluded to
is perfectly represented to the eye in the disturbed sub-
sidence of a precipitate, as described in art. 330, when
the fluid from which it subsides is warm, and losing heat
from its surface.

(336.) The sun's rays are the ultimate source of al-
most every motion which takes place on the surface of
the earth. By its heat are produced all winds, and those
disturbances in the electric equilibrium of the atmosphere
which give rise to the phenomena of terrestrial magnet-
ism. By their vivifying action vegetables are elaborated
from inorganic matter, and become, in their turn, the sup-
port of animals and of man, and the sources of those
great deposites of dynamical efficiency which are laid up
for human use in our coal strata. By them the waters
of the sea are made to circulate in vapour through the
ait, and irrigate the land, producing springs and rivers.
By them are produced all disturbances of the chymical
equilibrium of the elements of nature, which, by a series
of compositions and decompositions, give rise to new
products, and originate a transfer of materials. Even
the slow degradation of the solid constituents of the sur-
face, in which its chief geological changes consist, and
their diffusion among the waters of the ocean, are entirely
due to the abrasion of the wind and rain, and the alter-
nate action of the seasons ; and when we consider the
immense transfer of matter so produced, the increase of

202 A TREATISE ON ASTRONOMY. [cHAP. V.

pressure over large spaces in the bed of the ocean, and
diminution over corresponding portions of the land, we
are not at a loss to perceive how the elastic power of
subterranean fires, thus repressed on the one hand and
relieved on the other, may break forth in points when
the resistance is barely adequate to their retention, and
thus bring the phenomena of even volcanic activity under
the general law of solar influence.

(337.) The great mystery, however, is to conceive
how so enormous a conflagration (if such it be) can be
kept up. Every discovery in chymical science here
leaves us completely at a loss, or rather, seems to remove
farther the prospect of probable explanation. If conjec-
ture might be hazarded, we should look rather to the
known possibility of an indefinite generation of heat by
friction, or to its excitement by the electric discharge,
than to any actual combustion of ponderable fuel, whe-
ther solid or gaseous, for the origin of the solar radiation.*

* Electricity traversing excessively rarefied air or vapours, gives out
light, and, doubtless, also heat. May not a continual current of electric
matter be constantly circulating in the sun's immediate neiglibourhood,
or traversing the planetary spaces, and exciting, in the upper regions of
its atmosphere, those phenomena of which, on however diminutive a
scale, we have yet an unequivocal manifestation in our aurora borealis ?
The possible analogy of the solar light to that of the aurora has been
distinctly insisted on by my faiher, in his paper already cited. It would
be a highly curious subject of experimental inquiry, how far a mere re-
duplication of sheets of tlame, at a distance one behind the other (by
which their light might be brought to any required intensity), would com-
municate to the heat of the resulting compound ray the penetrating cha-
racter which distinguishes the solar calorific rays. We may also observe,
that the tranquillity of the sun's polar, as compared with its equatorial
regions (if its spots be really atmospheric), cannot be accounted for by its
rotation on its axis only, but 7nn.1t arise from some cause external t5 the
sun, as we see the belts of Jupiter and Saturn, and our trade-winds, aiise
from a cause, external to these planets, combining itself with iheir rota-
tion, which alone can produce no motions when once the form of equili.
brium is attained.

The prismatic analysis of the solar beam exhibits in the spectrum a
series of " fixed lines," totally vmlike those which Ijelong to the liglit of
any known terrestrial flame. This may hereafter lead us to a clearer
insight into its origin. But, before we can draw any conclusions from
such an indication, we must recollect, that previous to reaching us it has
undergone the whole absorptive action of our atmosphere, as well as of
the sun's. Of the latter we know nothing, and may conjecture every
thing ; but of the blue colour of the former we are sure ; and if this be
an inherent (;'. e. an absorptive) colour, tlie air must be expected to act
on the spectrum after the analogy of other coloured media, which often
(and especially light blue media) leave unabsorbed portions separateti by
dark intervals. It deserves inquiry, therefore, whether some or all the

CHAP. VI. J OF THE MOON. 203

CHAPTER VI.

Of the Moon— Its sidereal Period — Its apparent Diameter— Its ParallaS,
Distance, and real Diameter — First Approximation to its Orbit — Art
Ellipse about the Earth in the Focus— Its Eccentricity and Inclina-
tion—Motion of the Nodes of its Orbit— Occultations— Solar Eclipses
— Phases of the Moon — Its synodical Period — Lunar Eclipses —
Motion of the Apsides of its Orbit — Physical Constitution of the Moon
— Its Mountains — Atmosphere — Rotation on Axis — Libration — Ap-
pearance of the Earth from it.

(388.) The moon, like the sun, appears to advance
among the stars with a movement contrary to the general
diurnal motion of the heavens, but much more rapid, so
as to be very readily perceived (as Ave have before ob-
served) by a few hours' cursory attention on any moon-
light night. By this continual advance, which, though
sometimes quicker, sometimes slower, is never intermit-
ted or reversed, it makes the tour of the heavens in a
mean or average period of 27*^7^ 43"" ll'-5, returning,
in that time, to a position among the stars nearly coin-
cident with that it had before, and which would be ex-
actly so, but for causes presently to be stated.

(339.) The moon, then, like the sun, apparently de-
scribes an orbit round the earth, and this orbit cannot be
very different from a circle, because the apparent angular
diameter of the full moon is not liable to any gi-eat extent
of variation.

(340.) The distance of the moon from the earth is
concluded from its horizontal parallax, which may be found
either directly, by observations at remote geographical
stations, exactly similar to those described in art. 302,
in the case of the sun, or by means of the phenomena
called occultations (art. 346), from which also its appa-
rent diameter is most readily and correctly found. From
such observations it results that the mean or average dis-

fixed lines observed by WoUaston and Fraunhofer may not have their
origin in our own atmosphere. Experiments made on lofty mountains,
or the cars of balloons, on the one hand, and on the other with reflected
beams which have been made to traverse several miles of additional air
near the surface, would decide this point. The absorptive effect of the
sun's atmosphere, and possibly also of the medium surrounding it (what-
ever it be), which resists the motions of comets, cannot be thus eliminated.
— Author.

204 A TREATISE ON ASTRONOMY. [cHAP. Vl.

tance of the conlre of the moon from that of the earth
is 59*9643 of the earth's equatorial radii, or about
237,000 miles. This distance, great as it is, is little
more than one fourth of tlie diameter of the sun's body,
so that the globe of the sun would nearly twice include
the whole orbit of the moon ; a consideration wonderfully
calculated to raise our ideas of that stupendous lumi-
nary !

(341.) The distance of the moon's centre from an ob-
server at any station on the earth's surface, compared
with its apparent angular diameter as measured from that
station, will give its real or linear diameter. Now, the
former distance is easily calculated when the distance
from the earth's centre is known, and the apparent zenith
distance of the moon also determined by observation ;
for if we turn to the figure of art. 298, and suppose S the
moon, A the station, and C the earth's centre, the dis-
tance SC, and the earth's radius CA, two sides of the
triangle ACS are given, and the angle CAS, which is the
supplement of ZAS, the observed zenith distance, whence
it is easy to find AS, the moon's distance from A. From
such observations and calculations it results, that the
real diameter of the moon is 2160 miles, or about 0*2729
of that of the earth, whence it follows that the bulk of
the latter being considered as 1, that of the former will
be 0-0204, or about ^\.

(342.) By a series of observations, such as described
in art. 340, if continued during one or more revolutions
of the moon, its real distance may be ascertained at every
point of its orbit ; and if at the same time its apparent
places in the heavens be observed, and reduced by means
of its parallax to the earth's centre, their angular inter-
vals will become known, so that the path of the moon
may then be laid down on a chart supposed to represent
the plane in Avhich its orbit lies, just as was explained in
the case of the solar ellipse (art. 292). Now, when this
is done, it is found that, neglecting certain small (though
very perceptible) deviations (of which a satisfactory ac-
count will hereafter be rendered), the form of the appa-
rent orbit, like that of the sun, is elliptic, but consider-
ably more eccentric, the eccentricity amounting to 0-05484

CHAP. VI.] REVOLUTION OF THE MOOn's NODES. 205

of the mean distance, or the major semi-axis of the ellipse,
and the earth's centre being situated in its focus.

(343.) The plane in which this orbit lies is not the
ecliptic, however, but is inclined to it at an angle of 5°
8' 48", which is called the inclination of the lunar orbit,
and intersects it in two opposite points, which are called
its node — the ascending node being that in which the
moon passes from the southern side of the ecliptic to the
northern, and the descending the reverse. The points
of the orbit at which the moon is nearest to, and farthest
from, the earth, are called respectively its perigee and
apogee, and the line joining them and the earth the line
of apsides.

(344.) There are, however, several remarkable cir-
cumstances which interrupt the closeness of the analogy,
which cannot fail to strike the reader, between the mo-
tion of the moon around the earth, and of the earth round
the sun. In the latter case, the ellipse described remains,
during a great many revolutions, unaltered in its position
and dimensions ; or, at least, the changes which it under-
goes are not perceptible but in a course of veiy nice ob-
servations, which have disclosed, it is true, the existence
of " perturbations," but of so minute an order, that, in
ordinary parlance, and for common purposes, we may
leave them unconsidered. But this cannot be done in
the case of the moon. Even in a single revolution, its
deviation from a perfect ellipse is very sensible. It does
not return to the same exact position among the stars
from which it set out, thereby indicating a continual
change in the plane of its orbit. And, in effect, if we
trace by observation, from month to month, the point
where it traverses the ecliptic, we ^lall find that the nodes
of its orbit are in a continual state of retreat upon the
ecliptic. Suppose O to be the earth, and Ab ad that
portion of the plane of the ecliptic which is intersected
by the moon, in its alternate passages through it, from
south to north, and vice versa ; and let ABCDEF be a
portion of the moon's orbit, embracing a complete side-
real revolution. Suppose it to set out from the ascending
node, A ; then, if the orbit lay all in one plane, passing
through O, it would have a, the opposite point in the

S

206 A TREATISE ON ASTRONOMY. [cHAP. VI.

ecliptic, for its descending node ; after passing which, it
would again ascend at A. But, in fact, its real path car-

ries it not to a, but along a certain curve, ABC, to C aj
point in the ecliptic less than 180° distant from A ; so
that the angle AOC, or the arc of longitude described
between the ascending and the descending node, is some-
what less than 180°. It then pursues its course below
the ecliptic, along the curve CDE, and rises again above
it, not at the point c, diametrically opposite to C, but at
a point E, less advanced in longitude. On the whole,
then, the arc described in longitude between two conse-
cutive passages from south to north, through the plane
of the ecliptic, falls short of 360° by the angle AOE ;
or, in other words, the ascending node appears to have
retreated in one lunation, on the plane of the ecliptic by
that amount. To complete a sidereal revolution, then, it
must still go on to describe an arc, AF, on its orbit,
which will no longer, however, bring it exactly back to
A, but to a point somewhat above it, or having north lati'
tude.

(345.) The actual amount of this retreat of the moon's
node is about 3' 10"'64 joer diem, on an average, and in
a period of 6793"39 mean solar days, or about 18'6 years,
the ascending node is carried round in a direction con-
trary to the moon's motion in its orbit (or from east to
west) over a whole circumference of the ecliptic. Of
course, in the middle of this period the position of the
orbit must have been precisely reversed from what it Avas
at the beginning. Its apparent path, then, will lie among
totally different stars and constellations at different parts
of this period ; and, this kind of spiral revolution being
continually kept up, it will, at one time or other, cover
with its disc every point of the heavens within that

*►■

CHAP. VI.] ECLIPSES AND OCCtTLTATIONS. 207

limit of latitude or distance from the ecliptic which its
inclination permits ; that is to say, a belt or zone of the
heavens, of 10° 18' in breadth, having the ecliptic for its
middle line. Nevertheless, it still remains true that the
actual place of the moon, in consequence of this motion,
deviates in a single revolution very little from what it
would be were the nodes at rest. Supposing the moon
to set out from its node A, its latitude, when it comes to
F, having completed a revolution in longitude, will not
exceed 8' ; and it must be borne in mind that it is to ac-
count for, and represent geometrically, a deviation of this
small order, that the motion of the nodes is devised.

(346.) NoWs as the moon is at a very moderate dis-
tance from us (astronomically speaking), and is in fact
our nearest neighbour, while the sun and stars are in
comparison immensely beyond it, it must of necessity
happen, that at one time or other it must pass over and
occult or eclipse every star and planet within the zone
above described (and, as seen from the surface of earth,
even somewhat beyond it, by reason of parallax, M'hich
may throw it apparently nearly a degree either way
from its place as seen from the centre, according to the
observer's station). Nor is the sun itself exempt from
being thus hidden, whenever any part of the moon's
disc, in this her tortuous course, comes to overlap any
part of the space occupied in the heavens by.that lumi-
nary. On these occasions is exhibited the most striking
and impressive of all the occasional phenomena of astro-
nomy, an eclipse of the sim, in which a greater or less
portion, or even in some rare conjunctures the whole, of
its disc is obscured, and, as it were, obliterated, by the
superposition of that of the moon, which appears upon
it as a circularly-terminated black spot, producing a
temporary diminution of daylight, or even nocturnal
darkness, so that the stars appear as if at midnight. In
other cases, when, at the moment that the moon is cen-
trally superposed on the sun, it so happens that her dis-
tance from the earth is such as to render her angnilar
diameter less than the sun's, the very singular pheno-
menon of an annular solar eclipse takes place, when
the edge of tlie sun appears for a few minutes as a nar-.

208 A TREATISE ON ASTRONOiMY. [cHAP. VI.

row ring of light, projecting on all sides beyond the dark
circle occupied by the moon in its centre.

(347.) A solar eclipse can only happen when the sun
and moon are in conjunction, that is to say, have the
same, or nearly the same, position in the heavens, or the
same longitude. It will presently be seen that this con-
dition can only be fulfilled at the time of a new moon,
through it by no means follows, that at every conjunction
there must be an eclipse of the sun. If the lunar orbit
coincided with the ecliptic, this would be the case, but
as it is inclined to it at an angle of upwards of 5°, it is evi-
dent that the conjunction, or equality of longitudes, may
take place when the moon is in the part of her orbit too
remote from the ecliptic to permit the discs to meet and
overlap. It is easy, however, to assign the limits within
which an eclipse is possible. To this end we must con-
sider, that, by the effect of parallax, the moon's appa-
rent edge may be thrown in any direction, according to
a spectator's geographical station, by any amount not
exceeding the horizontal parallax. Now, this comes to
the same (so far as the possibility of an eclipse is con-
cerned) as if the apparent diameter of the moon, seen
from the earth's centre, were dilated by twice its hori-
zontal parallax ; for, if, when so dilated, it can touch or
overlap the sun, there inust be an eclipse at some part or
other of the earth's surface. If, then, at the moment of
the nearest conjunction, the geocentric distance of the
centres of the two luminaries do not exceed the sum of
their semidiameters and of the moon's horizontal paral-
lax, there will be an eclipse. This sum is, at its maxi-
mum, about 1° 34' 27". In the spherical triangle SNM,

then, in which S is the sun's centre, M the moon's, SN
the ecliptic, MN the moon's orbit, and N the node, we

CHAP. VI.] LIMITS OF A SOLAR ECLIPSE. 209

may suppose the angle NSM a right angle, SM = 1° 34'
27", and the angle MNS = 5° 8' 48", the mclmation of
the orbit. Hence we calculate SN, which comes out
16° 58'. If, then, at the moment of the new moon, the
moon's node is farther from the sun in longitude than
this limit, there can be no eclipse ; if within, there may,
and probably will, at some part or other of the earth.
To ascertain precisely whether there will or not, and,
if there be, how great will be the part eclipsed, the solar
and lunar tables must be consulted, the place of the node
and the semidiameters exactly ascertained, and the local
parallax, and apparent augmentation of the moon's dia-
meter due to the difference of her distance from the
observer and from the centre of the earth (which may
amount to a sixtieth part of her horizontal diameter),
determined ; after which it is easy, from the above con-
siderations, to calculate the amount overlapped of the
two discs, and their moment of contact.

(348,) The calculation of the occultation of a star
depends on similar considerations. An occultation is
possible, when the moon's course, as seen from the
earth's centre, carries her within a distance from the
star equal to the sum of her semidiameter and horizontal
parallax; and it tvill happen at any particular spot,
when her apparent path, as seen from that spot, carries
her centre within a distance equal to the sum of her
augmented semidiameter and actual parallax. The de-
tails of these calculations, which are somewhat trouble-
some, must be sought elsewhere.*

(349.) The phenomenon of a solar eclipse and of an
occultation are highly interesting and instructive in a
physical point of view. They teach us that the moon
is an opaque body, terminated by a real and sharply de-
fined surface intercepting light like a solid. They prove
to us, also, that at those times when we cannot see the
moon, she really exists, and pursues her course, and
that when we see her only as a crescent, however nar-
row, the whole globular body is there, filling up the de-
ficient outline, though unseen. For occultations take
place indifferently at the dark and bright, the visible and

* Woodhoase's Astronomy, vol. i. See also Trans. Ast. Soc. vol. 1. p. 325,

s2

210 A TREATISE ON ASTRONOMY. [cHAP. VI.

invisible oulline, whichever happens to be towards the
direction in which the moon is moving ; with this only
difference, that a star occulted by the bright limb, if the
phenomenon be watched with a telescope, gives notice,
by its gradual approach to tlie visible edge, when to ex-
pect its disappearance, while, if occulted at the dark
limb, if the moon, at least, be more than a few days
old, it is, as it were, extinguished in mid-air, without
notice or visible cause for its disappearance, which, as
it happens instantaneously, and without the slightest
previous diminution of its light, is always surprising ;
and, if the star be a large and bright one, even startling
from its suddenness. The reappearance of the star, too,
when the moon has passed over it, takes place in those
cases when the bright side of the moon is foremost, not
at the concave outline of the crescent, but at the invisible
outline of the complete circle, and is scarcely less sur-
prising, from its suddenness, than its disappearance in
the other case.*

(350.) The existence of the complete circle of the disc,
even when the moon is not full, does not, however, rest
only on the evidence of occultations and eclipses. It
may be seen, when the moon is crescent or waning, a few
days before and after the new moon, with the naked eye,
as a pale round body to which the crescent seems attach-
ed, and somewhat projecting beyond its outline (which
is an optical illusion arising from the greater intensity of
its light). The cause of this appearance will presently
be explained. Meanwhile the fact is sufficient to show

* There is an optical illusion of a very strange and unaccountable na-
ture which has often been remarked in occultations. The star appears
to advance actually upon and within the edge of the disc before it disap-
pears, and that sometimes to a considerable depth. I have never myself
witnessed this singular effect, but it rests on most unequivocal testimony.
I have called it an optical illusion ; but it is barely possible that a star
may shine on such occasions through deep fissures in the substance of
the moon. The occultations of close double stars ought to be narrowly
watched, to see whether both individuals are thus projected, as well as
for other purposes connected with their theory. I will only hint at one,
viz. that a double star, too close to be seen divided with any telescope,
may yet be detected to be double by the mode of its disappearance.
Should a considerable star, for instance, instead of undergoing instanta-
neous and complete extinction, go out by two distinct steps, following
close upon each other; first losing a portion, then the whole remainder
of its liglit, we may be sure it is a double star, though we camiot see the
individuals separately. — Author.

CHAP. VI, 3 PIIASr^S OF THE MOON. 211

that the moon is not inherently luminous like the sun,
but that her liglit is of an adventitious nature. And its
crescent form, increasing regularly from a narrow
semicircular line to a complete circular disc, corres-
ponds to the appearance a globe would present, one he-
misphere of which was black, the other wliite, when dif-
ferently turned towards the eye, so as to present a great-
er or less portion of eacli. The obvious conclusion from
this is, that the moon is such a globe, one half of Avhich
is brightened by the rays of some luminary sufficiently
distant to enlighten the complete hemisphere, and suffi-
ciently intense to give it the degree of splendour we see.
Now, the sun alone is competent to such an effect. Its
distance and light suffice ; and, moreover, it is invariably
observed that, when a crescent, the bright edge is towards
the sun, and that in proportion as the moon in her monthly
course becomes more and more distant from the sun, the
breadth of the crescent increases, and vice versa.

(351.) The sun's distance being 2.3984 radii of the
earth, and the moon's only 60, the former is nearly 400
times the latter. Lines, therefore, drawn from the sun
to every part of the moon's orbit may be regarded as par-
allel. Suppose, now, O to be the earth, ABCD, &c.

G

my ^^m

F

various positions of the moon in its orbit, and S the sun,
at the vast distance above stated ; as is shown, then, in
the figure, the hemisphere of the lunar globe turned to-
wards it (on the right) will be bright, the opposite dark,
wherever it may stand in its orbit. Now, in the position
A, when in conjunction with the sun, the dark part is

212 A TREATISE ON ASTRONOMY. [cHAP. VI.

entirely turned towards O, and the bright from it, Iir
this case, then, the moon is not seen, it is neiv moon.
When the moon has come to C, half the bright and half
the dark hemisphere are presented to O, and the same in
the opposite situation G : these are the first and third
quarters of the moon. Lastly, when at E, the whole
brigfht face is towax-ds the earth, the whole dark side from
it, and it is then seen wholly bright o\ full moon. In the
intermediate positions BDFH, the portions of the
bright face presented to O will be at first less than half
the visible surface, then greater, and finally less again,
till it vanishes altogether, as it comes round again to A.

(.352.) These monthly changes of appearance, or
phases, as they are called, arise, then, from the moon, an
opaque body, being illuminated on one side by the sun,
and reflecting from it, in all directions, a portion of the
light so i-eceived. Nor let it be thought surprising that
a solid substance thus illuminated should appear to shine
and again illuminate the earth. It is no more than a
white cloud does standing ofl' upon the clear blue sky.
By day, the moon can hardly be distinguished in bright-
ness from such a cloud ; and, in the dusk of evening,
clouds catching the last rays of the sun appear with a
dazzling splendour, not inferior to the seeming brightness
of the moon at night. That tlie earth sends also such a
light to the moon, orJy probably more powerful by rea-
son of its greater apparent size,* is agi'eeable to optical
principles, and explains the appearance of the dark por-
tion of the young moon completing its crescent (art. 350).
For, when the moon is nearly new to the earth, the lat-
ter (so to speak) is nearly full to the former ; it then illu'
minates its dark half by strong earth-light ^ and it is a
portion of this, reflected back again, which makes it visi-
ble to us in the twilight sky. As the moon gains age,
the earth ofl^ers it a less portion of its bright side, and the
phenomenon in question dies away.

(353.) The lunar month is determined by the recur-
rence of its phases ; it reckons from new moon to new

*The apparent diameter of the moon is 32' from the earth; that of the
earth seen from the moon is twice her horizontal parallax, or 1" 54'. The
apparent surfaces, therefore, are as (114)2 ; (32)2, or as 13 ; 1 nearly.

CHAP. VI. J SYNODICAL REVOLUTION OF TIH: MOOX. 218

moon ; that is, from leaving its conjunction with the sun
to its return to conjunction. If the sun stood still, like a
fixed star, the interval between two conjunctions would
be the same as the period of the moon's sidereal revolu-
tion (art. 338) ; but, as the sun apparently advances in
the heavens in the same direction with the moon, only
slower, the latter has more than a complete sidereal pe-
riod to perform to come up with the sun again, and will
require for it a longer time, which is the lunar month,
or, as it is generally termed in astronomy, a synodical
period. The difference is easily calculated by consider-
ing that the superfluous arc (whatever it be) is described
by the sun with his velocity of 0° -98565 per diem, in
the same time that the moon describes that arc yj/ws a
complete revolution, with her velocity of 13°"17640 joer
diem ; and, the times of description being identical, the
spaces are to each other in the proportion of the veloci-
ties.* From these data a slight knowledge of arithmetic
will suffice to derive the arc in question, and the time
of its description by the moon ; which, being the excess
of the synodic over the sidereal period, the former will
be had, and will appear to be 29'' 12'' 44"" 2^-87.

(354.) Supposing the position of the nodes of the
moon's orbit to permit it, when the moon stands at A
(or at the new moon), it will intercept a part or the
whole of the sun's rays, and cause a solar eclipse. On
the other hand, Avhen at E (or at the full moon), the
earth O will intercept the rays of the sun, and cast a
shadow on the moon, thereby causing a lunar eclipse.
And this is perfectly consonant to fact, such eclipses
never happening but at the exact time of the full moon.
But, what is still more remarkable, as confirmatory of
the position of the earth's sphericity, this shadow, which
we plainly see to enter upon, and, as it were, eat away
the disc of the moon, is always terminated by a circular
outline, though, from the greater size of the circle, it is

* Let V and w be the mean angular velocities, x the superfluous arc ;
thenV: w;; I+2:a;;and V — v.v.-.l .- a;, whence a; is found, and- =the

V

time of describing x, or the difference of the sidereal aad syiiodical peri-
ods. We shall have occasion for tliis again.

214 A TREATISE ON ASTRONOMY [cHAP. VI.

only partially seen at any one time. Now, a body
which always casts a circular shadow must itself be
spherical,

(355.) Eclipses of the sun are best understood by re-
garding the sun and moon as two independent luminaries,
each moving according to known laws, and viewed from
the earth ; but it is also instructive to consider eclipses
generally as arising from the shadow of one body thrown
on another by a luminary much larger than either. Sup-
pose, then, AB to represent the sun, and CD a spherical
body, whether earth or moon, illuminated by it. If we
join and prolong AC, BD ; since AB is greater than CD,
these lines will meet in a point E, more or less distant
from the body CD, according to its size, and within the
gpace CED (which represents a cone, since CD and AB

pre spheres), there will be a total shadow. This shadow
is called the umbra, and a spectator situated within it
can see no part of the sun's disc. Beyond the umbra
are two diverging spaces (or rather, a portion of a single
conical space, having K for its vertex), where if a
spectator be situated, as at M, he will see a portion only
(AONP) of the sun's surface, the rest (BONP) being ob-
scured by the earth. He will, therefore, receive only
partial sunshine ; and the more, the nearer he is to the
exterior borders of that cone which is called the penum^
bra. Beyond this he will see the whole sun, and be in
full illumination. All these circumstances may be per-;
fectly well shown by holding a small globe up in the.

CHAt>. Vl.J SYNODICAL REVOLUTION OF THE MOON. 218

sun, and receiving its shadow at different distances on a
sheet of paper.

(356.) In a lunar eclipse (represented in the upper
figure), the moon is seen to enter the penumbra first, and
by degrees, get involved in the umbra, the former sur-
rounding the latter like a haze. Owing to the great size
of the earth, the cone of its umbra always projects
far beyond the moon ; so that, if, at the time of the
eclipse, the moon's path be properly directed, it is sure
to pass through the umbra. This is not, however, the
case in solar eclipses. It so happens, from the adjust-
ment of the size and distance of the moon, that the ex-
tremity of her umbra always falls near the earth, but
sometimes attains and sometimes falls short of its surface.
In the former case (represented in the lower figure), a
black spot, surrounded by a fainter shadow, is formed,
beyond which there is no eclipse on any part of the
. earth, but within which there may be either a total or
partial one, as the spectator is within the umbra or
pemwibra. When the apex of the umbra falls on the
surface, the moon at that point will appear, for an in-
stant, to jzist cover the sun ; but, when it falls short,
there will be no total eclipse on any part of the earth ;
but a spectator, situated in or near the prolongation of
the axis of the cone, will see the whole of the moon on
the sun, although not large enough to cover it, i. e. he
will witness an annular eclipse.

(357.^ Owing to a remarkable enough adjustment of
the periods in which the moon's sy nodical revolution,
and that of her nodes, are performed, eclipses return aftef
a certain period, very nearly in the same order and of the
same magnitude. For 223 of the moon's mean synodi-
cal revolutions, or lunations, as they are called, will be
found to occupy 6585-32 days, and nineteen complete
synodical revolutions of the node to occupy 6585-78.
The difference in the mean position of the node, then, at
the beginning and end of 223 lunations, is nearly insen-
sible ; so that a recurrence of all eclipses within that in-
terval must take place. Accordingly this period of 223'
lunations, or eighteen years and ten days, is a very im-
portant one in the calculation of eclipses. It is supposed

216 A TREATISE ON ASTRONOMV. [cHAP. VI.

to have been known to the Chaldeans, under the name of
the saros ; the regular return of eclipses having been
known as a physical fact for ages before their exact the-
ory was understood. )

(358.) The commencement, duration, and magnitude
of a lunar eclipse are much more easily calculated than
those of a solar, being independent of the position of the
spectator on the earth's surface, and the same as if view-
ed from its centre. The common centre of the umbra
and penumbra lies always in the ecliptic, at a point oppo-
site to the sun, and the path described by the moon in pass-
ing through it is its true orbit, as it stands at the moment
of the full moon. In this orbit, its position, at every in-
stant, is known from the lunar tables and ephemeris ; and
all we have, therefore, to ascertain is, the moment ivhen
the distance between the moon's centre and the centre of
the shadow is exactly equal to the sum of the seinidiame-
ters of the moon and penumbra, or of the moon and
umbra, to know when it enters upon and leaves them re-
spectively.

(359.) The dimensions of the shadow, at the place
where it crosses the moon's path, require us to know
the distances of the sun and moon at the time. These
are variable ; but are calculated and set down, as well as
their semidiameters, for every day, in the epliemeris, so
that none of the data are wanting. The sun's distance is
easily calcidated from its elliptic orbit ; but the moon's
is a matter of more difficulty, for a reason we will now
explain.

(360.) The moon's orbit, as we have befoi'e hinted, is
not, strictly speaking, an ellipse returning into itself, by
reason of the variation of the plane in which it lies, and
the motion of its nodes. But even laying aside this con-
sideration, the axis of the ellipse is itself constantly
changing its direction in space, as has been already stated
of the solar ellipse, but much more rapidly ; making a
complete revolution, in the same direction with the moon's
own motion, in 3232*5753 mean solar days, or about
nine years, being about 3° of angular motion in a whole
revolution of the moon. This is the phenomenon known

CHAP. VI,] PHYSICAL CONDITION OF THE MOON. 317

by the name of the revohition of the moon's apsides. Its
cause will be hereafter explamed. Its immediate effect
is to produce a variation in the moon's distance from the
earth, which is not included in the laws of exact elliptic
motion. In a single revolution of the moon, this varia-
tion of distance is trifling ; but in the course of many it
becomes considerable, as is easily seen, if we consider
that in four years and a half the position of the axis will
be completely reversed, and the apogee of the moon will
occur where the perigee occurred before.

(361.) The best way to form a distinct conception of
the moon's motion is to regard it as describing an ellipse
about the earth in the focus, and, at the same time, to re-
gard this ellipse itself to be in a twofold state of revolu-
tion ; 1st, in its own plane, by a continual advance of its
axis in that plane ; and 2dly, by a continual tilting mo-
tion of the piano itself, exactly similar to, but much more
rapid than, that of the earth's equator produced by the
conical motion of its axis described in art. 266.

(362.) The physical constitution of the moon is better
known to us than that of any other heavenly body. By
the aid of telescopes, v/e discern inequalities in its sur-
face which can be no other than mountains and valleys — •
for this plain reason, that we see the shadows cast by thd
former in the exact proportion as to length which they
ought to have, when we take into account the inclination
of the sun's rays to that part of the moon's surface drl
which they stand. The convex outline of the limb turned
towards the sun is always circular, and very nearly
smooth ; but the opposite border of the enlightened partj
which (were the moon a perfect sphere) ought to be ari
exact and sharply defined ellipse, is always observed to
be extremely ragged, and indented with deep recessea
and prominent points. The mountains near this edgd
cast long black shadows, as they should evidently doj
when we consider that the sun is in the act of rising or
setting to the parts of the moon so circ^nnstanced. Bui
as the enlightened edge advances beyond them, i. e. as
the sun to them gains altitude, their shadows shorten 5
and at the full moon, when all the light falls in our lind

T

218 A TREATISE ON ASTRONOMY. (^CHAP. VI«

of sight, no shadows are seen on any part of her surface.
From micrometrical measures of the lengths of the sha-
dows of many of tlie more conspicuous mountains, taken
under the most favourable circumstances, the heights of
many of them have been calculated ; the highest being
about 1| English miles in perpendicular altitude. The
existence of such mountains is corroborated by their ap-
pearance as small points or islands of light beyond the
extreme edge of the enlightened part, which are their
tops catching the sunbeams before the intermediate
plain, and which, as the light advances, at length connect
themselves with it, and appear as prominences from the
general edge.

(363.) The generality of the lunar mountains present a
striking uniformity and singularity of aspect. They are
wonderfully numerous, occupying by far the larger por-
tion of the surface, and almost universally of an exactly
circular or cup-shaped form, foreshortened, however, into
ellipses towards the limb ; but the larger have for the
most part flat bottom.s within, from which rises centrally
a small, steep, conical hill. They offer, in short, in its
highest perfection, the true volcanic character, as it may
be seen in the crater of Vesuvius, and in a map of the
volcanic districts of the Campi Phlegr^ei* or the Puy de
Dome. And in some of the principal ones, decisive
marks of volcanic stratification, arising from successive
deposites of ejected matter, may be clearly traced with
powerful telescopes.! What is, moreover, extremely
singular in the geology of the moon is, that although no-
thing having the character of seas can be traced (for the
dusky spots which are commonly called seas, Avhcn
closely examined, present appearances incompatible with
the supposition of deep water), yet there are large re-
gions perfectly level, and apparently of a decided alluvial
character.

(304.) The moon has no clouds, nor any other indi-
cations of an atmosphere. Were there any, it could not
fail to be perceived in the occultations of stars and the
phenomena of solar eclipses. Hence its climate must

* See Breislak's map of the environs of Naples, and Desmarest's of
Auvergne.
t From ray (nvn o?J3ervation& — Author.

CHAP. VI.3 PHYSICAL CONDITION OF THE MOON. 219

be very extraordinary ; the alternation being that of un-
mitigated and burning sunshine, fiercer than an equatorial
noon, continued for a whole fortnight, and the keenest
severity of frost, far exceeding that of our polar Avinters,
for an equal time. Such a disposition of things must
produce a constant transfer of whatever moisture may
exist on its surface, from the point beneath the sun to
that opposite, by distillation in vacuo after the manner
of the little instrument called a cryophoros. The con-
sequence must be absolute aridity below the vertical sun,
constant accretion of hoar frost in the opposite region,
and, perhaps, a narrow zone of running water at the
borders of the enlightened hemisphere. It is possible,
then, that evaporation on the one hand, and condensation
on the other, may to a certain extent preserve an equili-
brium of temperature, and mitigate the extreme severity
of both climates.

(365.) A circle of one second in diameter, as seen
from the earth, on the surface of the moon, contains
about a square mile. Telescopes, therefore, must yet be
greatly improved, before we could expect to see signs of
inhabitants, as manifested by edifices or by changes on
the surface of the soil. It should, however, be observed,
that, owing to the small density of the materials of the
moon, and the comparatively feeble gravitation of bodies
on her surface, muscular force would there go six times
as far in overcoming the weight of materials as on the
earth. Owing to tlie want of air, however, it seems im-
possible that any form of life analogous to those on earth
can subsist there. No appearance indicating vegetation,
or the slightest variation of surface which can fairly be
ascribed to change of season, can any where be discerned.

(366.) The lunar summer and winter arise, in fact,
from the rotation of the moon on its own axis, the period
of which rotation is exactly equal to its sidereal revolu-
tion about the earth, and is performed in a plane 1° 30'
11" inclined to the ecliptic, and therefore nearly coinci-
dent with her own orbit. This is the cause why we al-
ways see the same face of the moon, and have no know-
ledge of the other side. This remarkable coincidence
of two perio<ls, which at first sight would seem perfectly

220 A TREATISE ON ASTRONOMY. [cHAP. VI,

distinct, is said to be a consequence of the general laws
to be explained hereafter.

(367.) The moon's rotation on her axis is uniform ;
but since her motion in her orbit (like that of the sun) is
not so, Ave are enabled to look a few degrees round the
equatorial parts of her visible border, on the eastern or
western side, according to circumstances ; or, in other
words, tlie line joining the centres of the earth and moon
fluctuates a little in its position, from its mean or average
intersection Avith her surface, to the east or westward.
And, moreover, since the axis about which she revolves
is not exactly perpendicular to her orbit, her poles come
alternately into view for a small space at the edges of her
disc. These phenomena are knov/n by the name of li-
prutions. In consequence of these two distinct kinds of
libration, the same identical point of the moon's surface
is not always the centre of her disc, and we therefore get
sight of a zone of a few degrees in breadth on all sides
of the border, beyond an exact hemisphere.

(368.) if there be inhabitants in the moon, the earth
^aist present to them the extraordinary appearance of a
piioon cf nearly 2° in diameter, exhibiting the same phases
fts we see the moon to do, but hnmoveably fixed in their
sky (or, at least, changing its apparent place only by the
small amount of the libration), while the stars must seem
^o pass slowly beside and behind it. It v/ill appear
clouded with variable spots, and belted with equatorial
^nd tropical zones corresponding to our trade-winds ; and
it may be doubted whether, in their perpetual change, the
outlines of our contineiits and seas can ever be clearly
discerned^

CHAP. VII. J ON TERUESTRIAI. GRAVITY. 221

CHAPTER VII.

Of terrestrial Gravity— Of the Law of universal Gravitation— Paths of
Projectiles ; apparent, real — ^The Moon retained in her Orbit by Gravity
— Its Law of Diminution— Laws of elliptic Motion— Orbit of the Earth
round the Sun in accordance with these Laws — Masses of the Earth
anil Sun compared— Density of the Sun— Force of Gravity at its Sur-
face— Disturbing Effect of tlie Sun on the Moon's Motion.

(369.) The reader has now been made acquainted with
the chief phenomena of the motions of tlie earth in its
orbit round the sun, and of the moon about the earth.
We come next to speak of the physical cause which
maintains and perpetuates these motions, and causes the
massive bodies so revolving to deviate continually from
the directions they would naturally seek to follow, in
pursuance of the first law of motion,* and bend their
courses into curves concave to their centres.

(370.) Whatever attempts may have been made by
metaphysical writers to reason away the connexion of
cause and effect, and fritter it down into the unsatisfacto-
ry relation of habitual sequence,! it is certain that the
conception of some more real and intimate connexion is
quite as strongly impressed upon the human mind as that
of the existence of an external world, — the vindication
of wliose reality has (strange to say) been regarded as
an achievement of no common merit in the annals of this
branch of philosophy. It is our own immediate con-
sciousness of effort, when Ave exert force to put matter
in motion, or to oppose and neutralize force, which gives
us this internal conviction of poiver and causation so far
as it refers to the material world, and compels us to be-
lieve that whenever we see material objects put in motion

* See Cab. Cyc. Mechanics, chap. iii.

t See Brown " On Cause and Effect," — a work of great acuteness and
subtlety of reasoning on some points, but in which the wliole train of ar-
ginnent is vitiated by one enormous oversight ; the omission, namely, of
a. dislinct and immediate personal consciovsriess of causation in his enu-
meration of that sequence of evendi, by wliicli the volition of the mind is
made to terminate in the motion of material objects. I mean the con-
sciousness of effort, as a thing entirely distinct from mere desire or volition
on the one hand, and from mere spasmodic contraction of muscles on the
other. Brown, 3d edit Ediii. 1818, p. il.— Author.

T 2

222 A ITvKATISE ON ASTRONOMY. [cHAP. VII..

from a state of rest, or deflected from their rectilinear
paths, and changed in their velocities if already in motion,
it is in consequence of such an effort someAo?/? exerted,
i^hough not accompanied with our consciousness. That
such an effort should be exerted with success tlirough an
interposed space, is no more difficult to conceive than
that our hand should communicate motion to a stone,
\yith Avhich it is demonstrubly not in contact.

(.371.) All bodies with which we are acquainted, when
raised into the air and quietly abandoned, descend to the
(^ai'th's surface in lines perpendicular to it. They are
therefore urged thereto by a force or eflbrt, the direct or
indirect result of a consciousness and a loill existing
somewhere, though beyond our pov/er to trace, which
force Ave term gravity ; and whose tendency or direction,
^s imiversal experience teaches, is towards the earth's
centre ; or rather, to speak strictly, with reference to its
spheroidal figure, perpendicular to the surface of still
water. But if we cast a body obliquely into tlie air,
^his tendency, though not extinguished or diminished, ia
materially modified in its ultimate effect. The upward
ijmpetus we give the stone is, it is true, after a time de^
stroyed, and a downward one communicated to it, which
ultimately brings it to the surface, where it is opposed in
its further progress, and brought to rest. But all the
while it has been continually deflected or bent aside from
its. rectilinear progress, and made to describe a curved
line concave to the earth's centre ; and having a highest
■pointy vertex, or apogee, just as the moon has in its orbit,
where the direction of its motion is perpendicular to the
radius.

(372.) When the stone which we fling obliquely up-
wards meets and is stopped in its descent by the earth's
surface, its motion is not towards the centre, but inclined
to the earth's radius at the same angle as when it quitted
our hand. As we are sure that, if not stopped by the
resistance of the earth, it woidd continue to descend, and
that obliquehj, what presumption, we may ask, is there
that it would ever reach the centre, to which its motion,
in no part of its visible course, was ever directed ? What
reason have we to believe that it might not rather circii-*

OHAP. vn.J MOTION OF A PROJECTILE. 223

late round it, as the moon does round the earth, returning
again to the point it set out from, after completing an
elliptic orbit of which the centre occupies the lower
focus ? And if so, is it not reasonable to imagine that the
same force of gi-avity may (since we know that it is ex-
erted at all accessible heights above the surface, and even
in the highest regions of the atmosphere) extend as far
as 60 radii of the earth, or to the moon ? and may not
this be the power — for some power there must be —
which deflects her at every instant from the tangent of
her orbit, and keeps her in the elliptic path which expe-^
rience teaches us she actually pursues ?

(373.) If a stone be whirled round at the end of a
string, it will stretch the string by a centrifugal force,*
which, if the speed of rotation be sufficiently increased,
will at length break the string, and let the stone escape.
However strong the string, it may, by a sufficient rotatory
velocity of the stone-, be brought to the utmost tension it
will bear without breaking ; and if Ave know Avhat weight
it is capable of carrying, the velocity necessary for this
purpose is easily calculated. Suppose, now, a string to
connect the earth's centre, with a weight at its surface,
whose strength should be just sufficient to sustain that
weight suspended from it. Let us, however, for a mo-
ment imagine gravity to have no existence, and that the
weight is made to revolve with the limiting velociti/
which that string can barely counteract : then will its
tension be just equal to the weight of the revolving body ;
and any power which should continually urge the body
towards the centre with a force equal to its weight would
perform the office, and might supply the place of the
string, if divided. Divide it, then, and in its place let
gravity act, and the body will circulate as before ; its ten-
dency to the centre, or its iveight, being just balanced by
its centrifugal force. Knowing the radius of the earth,
we can calculate the periodical time in which a body so
balanced must circulate to keep it up ; and this appears
to be I'' 23™ 22^

(374.) If we make the same calculation for a body at
the distance of the moon, supposing its iveight or gra'>^
* See Cab. Cyc. Mechanics, chap. viii.

224 A TREATISE ON AfiTRONOMY. [cHAP. VII.

vity the same as at the earth'' s surface, we sliall find the
period required to be 10'' 45™ 30". The actual period of
the moon's revohition, however, is 27'^ 7'' 43™ ; and hence
it is clear that the moon's velocity is not nearly sufficient
to sustain it against such a power, supposing it to revolve
in a circle, or neglecting (for the present) the slight ellip-
ticity of its orbit. In order that a body at the distance
of the moon (or the moon itself) should be capable of
keeping its distance from the earth by the outward effort
of its centrifugal force, while yet its time of revolution
should be what the moon's actually is, it will appear (on
executing the calculation from the principles laid down
in Cab. Cyc. Mechanics) that gravity, instead of being
as intense as at the surface, would require to be very
nearly 3600 times less energetic ; or, in other words,
that its intensity is so enfeebled by the remoteness
of the body on which it acts, as to be capable of
producing in it, in the same time, only -^ oVo'^^ P^^'*- ^^
the motion which it Avould impart to the same mass
of matter at the earth's surface.

(375.) The distance of the moon from the earth's
centre is somewhat less than sixty times the distance
from the centre to the surface, and 3600 : 1 : : 60^ : P;
so that the proportion in which we must admit the earth's
gravity to be enfeebled at the moon's distance, if it be
really the force v/hieh- retains the moon in her orbit, must
be (at least in this particular instance) that of the squares
of the distances at which it is compared. Now, in such
a diminution of energy with increase of distance, there
is nothing prima facie inadmissible. Emanations from
a centre, such as light and heat, do really diminish in in-
tensity by increase of distance, and in this identical pro-
portion ; and though we cannot certainly argue much
from this analogy, yet v/e do see that the power of mag-
netic and electric attractions and repulsions is actually
enfeebled by distance, and much more rapidly than in
the simple proportion of the increased distances. The
argument, therefore, stands thus : — On the one hand,
gravity is a real power, of whose agency we have daily
experience. We know that it extends to the greatest ac-
cessible heights, and far beyond ; and we see no reason

CHAP. VII. 3 ATTRACTION OF SPHERES. 22S

for drawing a line at any particular height, and there as-
serting that it must cease entirely ; though we have ana-
logies to lead us to suppose its energy may diminish
rapidly as we ascend to great heights from the surface,
such as that of the moon. On the other hand, we are
sure the moon is urged towards the earth by some power
which retains her in her orbit, and that the intensity of
this power is such as would correspond to a diminished
gravity, in the proportion — otherwise not improbable —
of the squares of the distances. If gravity be not that
power, there must exist some other ; and, besides this,
gravity must cease at some inferior level, or the nature
of the^ moon must be different from that of ponderable
matter ; — for if not, it would be urged by both powers,
and' therefore too much urged, and forced inwards from
her path.

(376.) It is on such an argument that Newton is un-
derstood to have rested, in the first instance, and provi-
sionally, his law of universal gravitation, which may be
thus abstractly stated : — " Every particle of matter in
the universe attracts every other particle, with a force
directly proportioned to the mass of the attracting par-
ticle, and inversely to the square of the distance between
them." In this abstract and general form, however, the
proposition is not applicable to the case before us. The
earth and moon are not mere particles, but great spherical
bodies, and to such the general law does not immediately
apply ; and, before we can make it applicable, it becomes
necessary to inquire what will be the force with which a
congeries of particles, constituting a solid mass of any as-
signed fioairc, will attract another sucli collection of mate-.
rial atoms. This problem is one purely dynamical, and, m
its general form, is of extreme difficulty. Fortunately,
however, for human knowledge, when the attracting and
attracted bodies are spheres, it admits of an easy and di-
rect solution. Newton himself has shown [Princip.
b. i. prop. 75) that, in that case, the attraction is pre-
cisely the same as if the v/hole matter of each sphere
were collected into its centre, and the spheres were
single particles there placed ; so that, in this case, the
general law applies in ita strict wording. The effect of

226 A TREATISE ON ASTRONOMY. [cHAP. VII.

the trifling deviation of the earth from a spherical form
is of too minute an order to need attention at present.
It is, however, perceptible, and may be hereafter noticed.

(377.) The next step in the Newtonian ai-gument is
one which divests the law of gravitation of its provisional
character, as derived from a loose and superficial consi-
deration of the lunar orbit as a circle described with an
average or mean velocity, and elevates it to the rank of
a general and primordial relation, by proving its applica-
bility to the state of existing nature in all its detail of
circumstances. This step consists in demonstrating, as
he has done* (Princip. i. 17, i. 75), that, under the in-
fluence of such an attractive force mutually urging two
spherical gravitating bodies towards each other, they
will each, when moving in each other's neighbourheod,
be deflected into an orbit concave towards the other, and
describe, one about the other regarded as fixed, or both
round their common centre of gravity, curves whose
forms are limited to those figures known in geometry by
the general name of conic sections. It will depend, he
shows, in any assigned case, upon the particular circum-
stances of velocity, distance, and direction, which of
these curves shall be described, — whether an ellipse, a
circle, a parabola, or an hyperbola ; but one or other it
must be ; and any one of any degree of eccentricity it
77iay be, according to the circumstances of the case ; and,
in all cases, the point to which the motion is referred,
whether it be the centre of one of the spheres, or their
common centre of gravity, will of necessity be the focus
of the conic section described. He shows, furthermore
{Princip. i. 1), that in every case, the angular velocity
•with which the line joining their centres moves, must be
inversely proportional to the squai'e of their mutual dis-
tance, and that equal areas of the curves described will
be swept over by their line of junction in equal times.

(378.) All this is in conformity with what we have
stated of the solar and lunar movements. Their orbits

* We refer for these fundamental propositions, as a point of duty, to
the immortal work in which they were first propounded. It is impossi-
ble for us in this volume to go into these investigations : even did our
limits permit, it would be utterly inconsistent with our plan; a general
idea, however, of their conduct will be given in the liext chapter,

CHAP. VII.] MASS OF THE SUN. 22'7

are ellipses, but of different degrees of eccentricity ; and
this circumstance already indicates the general applica-
bility of the principles in question.

(379.) But here we have already, by a natural and
ready implication (such is always the progress of gene-
ralization), taken a further and most important step, al-
most unperceived. We have extended the action of
gravity to the case of the earth and sun, to a distance
immensely greater than tliat of the moon, and to a body
apparently quite of a different nature from either. Are
we justified in this ? or, at all events, are there no modi-
fications introduced by the change of data, if not into
the general expression, at least into the particular inter-
pretation, of the law of gravitation ? Now, the moment
we come to numbers, an obvious incongruity strikes us.
When we calculate, as above, from the known distance
of the sun (art. 304), and from the period in which the
earth circulates about it (art. 327), what must be the cen-
trifugal force of the latter by which the sun's attraction
is balanced (and which, therefore, becomes an exact
measure of the sun's attractive energy as exerted on the
earth), we find it to be immensely greater than would
suffice to counteract the eartli's attraction on an equal
body at that distance — greater in the high proportion of
35493G to 1. It is clear, then, that if the earth be re-
tained in its orbit about the sun by solar attraction, con-
formable in its rate of diminution with the general law,
this force must be no less than 354936 times more in-
tense than what the earth would be capable of exerting,
caeferis paribus, at an equal distance.

(380.) What, then, are we to understand from this
result ? Simply this, — that the sun attracts as a collec-
tion of 354936 earths occupying its place would do, or,
in other words, that the sun contains 354936 times the
mass or quantity of ponderable matter that the earth con-
sists of. Nor let this conclusion startle us. We have
only to recall what has been already shown in art. 305,
of the gigantic dimensions of this magnificent body, to
perceive that, in assigning to it so vast a mass, we are
not outstepping a reasonable proportion. In fact, when
we come to compare its mas3 with its bulk, we find its

^28 A TREATISE ON ASTRONOMY. [cHAP. VJl.

density* to be less than that of the earth, bein^ no more
than 0-2513. So that it must consist, in reality, of far
lighter materials, especially when we consider the force
under which its central parts must be condensed. This
consideration renders it highly probable that an intense
heat prevails in its interior, by which its elasticity is re-
inforced, and rendered capable of resisting this almost
inconceivable pressure without collapsing into smaller
dimensions.

(381.) This will be more distinctly appreciated, if wd
estimate, as we are now prepared to do, the intensity of
gravity at the sun's surface.

The attraction of a sphere being the same (art. 370)
as if its whole mass were collected in its centre, will, of
course, be proportional to the mass directly, and the
square of the distance inversely ; and, in this case, the
distance is the radius of the sphere. Hence Ave con-
clude,! that the intensities of solar and terrestrial gravity
at the surfaces of the two globes are in the proportions
of 27*9 to 1. A pound of terrestrial matter at the sun's
surface, then, would exert a pressure equal to what 27*9
such pounds would do at the earth's. An ordinary man,
for example, would not only be unable to sustain his own
weight on the sun, but would literally be crushed to
atoms under the load.|

(382.) Henceforward, then, we must consent to dis-
miss all idea of the earth's immobility, and transfer that
attribute to the sun, whose ponderous mass is calculated
to exhaust the feeble attractions of such comparative
atoms as the earth and moon, without being perceptibly
dragged from its place. Their centre of gravity lies, as
we have already hinted, almost close to the centre of
the solar globe, at an interval quite imperceptible from
our distance ; and whether Ave regard the earth's orbit as
being performed about the one or the other centre makes

* The density of a material body is aa the mass directly, and the
volume inversely : hence density of Q : density of @ : : ^~t ■ 1 .'
0-2543:1.

t Solar gravity : terrestrial : ■■ i^-^j : -^^-2 : : 279 : 1 ; the respec
tive radii of the sun and earth beijig 410000, and 4000 miles.

\ A mass weighing 12 stone or 170 lbs. on the eorth, would produce a
preesujfe of 40O0 lbs. on tJie aun.

CHAP. VII.] DISTURBANCE OF THE MOON's ORBIT. 239

no appreciable difference in any one phenomenon of
astronomy.

(383.) It is in consequence of the mutual gravitation
of all the several parts of matter, which the Newtonian
law supposes, that the earth and moon, Avhile in the act
of revolving, monthly, in their mutual orbits about their
common centre of gravity, yet continue to circulate,
without parting company, in a greater annual orbit round
the sun. We may conceive this motion by connecting
two unequal balls by a stick, which, at their centre of
gravity, is tied by a long string, and whirled round.
Their joint systems will circulate as one body about the
common centre to which the string is attached, while yet
they may go on circulating round each other in subor-
dinate gyrations, as if the stick were quite free from any
such tie, and merely hurled through the air. If the earth
alone, and not the moon, gravitated to the sun, it would
be dragged away, and leave the moon behind — and vice
versa; but, acting on both, they continue together under
its attraction, just as the loose parts of the earth's sur-
face continue to rest upon it. It is, then, in strictness,
not the earth or the moon which describes an ellipse
around the sun, but their common centre of gi-avity. The
effect is to produce a small, but very perceptible, monthly
equation in the sun's apparent motion as seen from the
earth, which is always taken into account in calculating
the sun's place.

(384.) And here, i. e. in the attraction of the sun, we
have the key to all those differences from an exact
elliptic movement of the moon in her monthly orbit,
which we have already noticed (arts. 344. 360), viz.
to the retrograde revolution of her nodes ; to the direct cir-
culation of the axis of her ellipse ; and to all the other
deviations from the laws of elliptic motion at which we
have further hinted. If the moon simply revolved about
the earth under the influence of its gravity, none of these
phenomena would take place. Its orbit would be a per-
fect ellipse, returning into itself, and always lying in one
and the same plane : that it is not so, is a proof that
some cause disturbs it, and interferes with the earth's
attraction ; and this cause is no other than the sun's at-

U

230 A TREATISE ON ASTRONOMY. [cHAP. Vlli

traction — or rather, that part of it which is not equally
exerted on the earth.

(385.) Suppose two stones, side by side, or otherwise
situated with respect to eacli other, to be let fall together ;
then, as gravity accelerates them equally, they will re-
tain their relative positions, and fall together as if they
formed one mass. But suppose gravity to be rather
more intensely exerted on one than the other ; then
would that one l)e rather more accelerated in its fall, and
would gradually leave the other ; and thus a relative
motion between them would arise from the difference of
action, however slight.

(386.) The sun is about 400 times more remote than
the moon ; and, in consequence, while the moon de-
scribes her monthly orbit round the earth, her distance
from the sun is alternately -j^^oth part greater and as
much less than the earth's. Small as this is, it is yet
sufhcient to produce a perceptible excess of attractive
tendency of the moon towards the sun, above that of the

XqiVL

S

earth when in the nearer point of her orbit, M, and a
corresponding defect on the opposite part, N ; and, in
the intermediate positions, not only will a difference of
forces subsist, but a difference of directions also ; since,
hoAvever small the lunar orbit MN, it is not a point, and,
therefore, the lines drawn from the sun S to its several
parts cannot be regarded as strictly parallel. If, as we
have already seen, the force of the sun were equally ex-
erted, and in parallel directions on both, no disturbance
of their relative situations would take place ; but from
the non-verification of these conditions arises a disturb-
ing force, oblique to the line joining the moon and earth,
which in some situations acts to accelerate, in others to
retard, her elliptic orbitual motion ; in some to draw the
earth from the moon, in others the moon from the earth.
Again, the lunar orbit, though very nearly, is yet not
quite coincident witli the plane of the ecliptic ; and hence
the action of the sun, which is very nearly parallel to the
last mentioned plane, tends to draw her somewhat out

CHAP, vni.] SOLAR SYSTEM. 231

of the plane of her orbit, and does actually do so — pro-
ducing the revolution of her nodes, and other phenomena
less striking. We are not yet prepared to go into the
suhjeci o( these perturbations, as they are called; but
they are introduced to the reader's notice as early as
possible, for the purpose of reassuring his mind, should
doubts have arisen as to the logical correctness of our
argument, in consequence of our temporary neglect of
them while working our way upward to the law of
gravity from a general consideration of the moon's orbit.

CHAPTER VIII.

OF THE SOLAR SYSTEM.

Apparent Motions of the Planets — ^Their Stations and Retrogradations —
The Sua their natural Centre of Motion — Inferior Planets — Tlieir
Phases, Periods, &c. — Dimensions and Form of their Orbits — Transits
across the Sun — Superior Planets — Their Distances, Periods, &c. —
Kepler's Laws and their Interpretation — Elliptic Elements of a Planet's
Orbit — Its heliocentric and geocentric Place — Bode's Law of planetary
Distances — The four iiltra-zodaical Planets — Physical Peculiarities ob-
servable in each of the Planets.

(387.) The sun and moon are not the only celestial
objects which appear to have a motion independent of
that by which the great constellation of the heavens is daily
carried round the earth. Among the stars there are seve-
ral,— and those among the brightes.t and most conspi-
cuous,— which, when attentively watched from night to
night, are found to change their relative situations among
the rest; some rapidly, others inuch more slowly. These
are called planets. Foitr of them — Venus, Mars, Ju-
piter, and Saturn — are remarkably large and brilliant ;
another. Mercury, is also visible to the naked eye as a
large star, but, for a reason which will presently appear,
is seldom conspicuous ; a fifth, Uranus, is barely dis-
cernible without a telescope ; and four others — Ceres,
Pallas, Vesta, and Juno — are never visible to the naked
eye. Besides these ten, others yet undiscovered may
exist ; and it is extremely probable that such is the case,
--e-tilie multitude of telescopic stars being so great that

233 A TREATISE ON A6TR0N0MY. [cHAP. Vlll.

only a small fraction of their number has been sufficiently
noticed to ascertain whether they retain the same places
or not, and the five last-mentioned planets having all been
discovered within half a century from the present time,

(388.) The apparent motions of the planets are much
more irregular than those of the sun or moon. Generally
speaking, and comparing their places at distant times,
they all advance, though with very different average or
7nean velocities, in the same direction as those lumina-
ries, i. e. in opposition to the apparent diurnal motion, or
from west to east : all of them make the entire tour of
the heavens, though under very different circumstances ;
and all of them, with the exception of the four telescopic
planets, — Ceres, Pallas, Juno, and Vesta (which may
therefore be termed ultra-zodiacal), — are confined in
their visible paths within very narrow limits on either
side the ecliptic, and perform their movements within
that zone of the heavens we have called above the Zo-
diac (art. 254).

(389.) The obvious conclusion from this is, that
whatever be, otherwise, the nature and law of their mo-
tions, they are all performed nearly in the plane of the
ecliptic, — that plane, namely, in which our own motion
about the sun is performed. Hence it follows, that we
see their evolutions, not in plan, but in section; their
real angular movements and linear distances being all
foreshortened and confounded undistinguishably, while
only their deviations from the ecliptic appear of their
natural magnitude, undiminished by the effect of per-
spective.

(390.) The apparent motions of the sun and moon,
though not uniform, do not deviate very greatly from
uniformity ; a moderate acceleration and retardation,
accountable for by the ellipticity of their orbits, being all
that is remarked. But the case is widely different with
the planets : sometimes they advance rapidly ; then re-
lax in their apparent speed — come to a momentary stop ;
and then actually reverse their motion, and run back upon
their former course, with a rapidity at first increasing,
then diminishing, till the reversed or retrograde motion
ceases altogether. Another station, or moment of ftp-

CHAP. VIII. 3 NODES OF A PLAXEt's ORBIT. 233

parent rest or indecision, now takes place ; after which
the movement is ajjain reversed, and resumes its orijjinal
direct character. On the whble, however, the amount
of direct motion more than compensates the retrograde ;
and by the excess of the former over tlie latter, the gra-
dual advance of the planet from west to east is main-
tained. Thus, supposing the zodiac to be unfolded into
a plane surface (or repi'esented as in Mercator's projec-
tion, art, 234, taking the ecliptic EC for its ground line),
the track of a planet, when mapped down by observation

from day to day, will offer the appearance PQRS, &c. ;
the motion from P to Q being direct, at Q stationary,
from Q to R retrograde, at R again stationary, from R
to S direct, and so on.

(391.) In the midst of the irregularity and fluctuation
of this motion, one remarkable feature of uniformity is
observed. Whenever the planet crosses the ecliptic, as
at N in the figure, it is said (like the moon) to be in its
node ; and as the earth necessarily lies in the plane of
the ecliptic, the planet cannot be apparently or iirano-
graphkally situated in the celestial circle so called, with-
out being really and locally situated in that plane. The
visible passage of a planet through its notle, then, is a
phenomenon indicative of a circumstance in its real mo-
tion quite independent of the station from which we view
it. Now, it is easy to ascertain, by observation, when a
planet passes from the north to the south side of the
ecliptic : we have only to convert its right ascensions
and declinations into longitudes and latitudes, and the
change from north to south latitude on two successive
days will advertise us on what day the transition took
place ; while a simple proportion, grounded on the ob-
served state of its motion in latitude in the interval,
will suffice to fix the precise hour and minute of its ar-
rival on the ecliptic. Now, this being done for several
transitions from side to side of the ecliptic, and their

v2

234 A TREATISE ON ASTRONOMY. [cHAP. VIII.

dates thereby fixed, we find, universally, that the interval
of time elapsing between the successive passages of each
planet through the same node (whether it be the ascend-
ing or the descending) is always alike, whether the planet
at the moment of such passage be direct or retrograde,
swift or slow, in its apparent movement.

(392.) Here, then, we have a circumstance wliich,
while it shows that the motions of the planets are in fact
subject to certain laws and fixed periods, may lead us
very naturally to suspect that the apparent irregularities
and complexities of their movements may be owing to
our not seeing them from their natural centre (art. 316),
and from our mixing up with their own proper motions
movements of a parallactic kind, due to our own change
of place, in virtue of the orbitual motion of the earth
about the sun.

(393.) If we abandon the earth as a centre of the pla-
netary motions, it cannot admit of a moment's hesitation
where we should place that centre with the greatest pro-
bability of truth. It must surely be the sun Avhich is
entitled to the first trial, as a station to which to refer
them. If it be not connected with them by any physical
relation, it at least possesses the advantage, which the
earth does not, of comparative immobility. But after
what has been shown in art. 380, of the immense mass
of that luminary, and of the office it performs to us as a
quiescent centre of our orbitual motion, nothing can be
more natural than to suppose it may perform the same
to other globes which, like the earth, may be revolving
round it ; and these globes may be visible to us by its
light reflected from them, as the moon is. Now there
are many facts which give a strong support to the idea
that the planets are in this predicament.

(394.) In the first place, the planets really are great
globes, of a size commensurate with the earth, and seve-
ral of them much greater. When examined through
powerful telescopes, they are seen to be round bodies, of
sensible and even of considerable apparent diameter, and
ofiering distinct and characteristic peculiarities,, which
show them to be solid masses, each possessing its indi-
vidual structure and mechanism ; and that, in one in-

CHAP. Vlll.] APPARENT DIAMETERS OF THE PLANETS. 235

stance at least, an exceedingly artificial and complex one.
(See the representations of Jupiter, Saturn, and Mars,
in plate I.) Tluit their distances from us are great,
much greater than that of the moon, and some of them
even greater than that of the sun, we infer from the
smallness of their diurnal parallax, which, even for the
nearest of them, when most favourably situated, docs
not exceed a few seconds, and for the more remote ones
is almost imperceptible. From the comparison of the
diurnal parallax of a celestial body, with its apparent
semidiameter, we can at once estimate its real size. For
the parallax is, in fact, nothing else than the apparent se-
midiameter of the earth as seen from the body in ques-
tion (art. 298, et seq.); and, the intervening distance
being the same, the real diameters must be to each other
in the proportion of the apparent ones. Without going
into particulars, it will suffice to state it as a general re-
sult of that comparison, that the planets are all of them
incomparably smaller than the sun, but some of them as
large as the earth, and others much greater.

(395.) The next fact respecting them is, that their
distances from us, as estimated from the measurement
of their angular diameters, are in a continual state of
change, periodically increasing and decreasing within
certain limits, but by no means corresponding with the
supposition of regular circular or elliptic orbits described
by them about the earth as a centre or focus, but main-
taining a constant and obvious relation to their apparent
angular distances or elongations from the sun. For ex-
ample ; the apparent diameter of Mars is greater when
in opposition (as it is called) to the sun, i. e. when in
the opposite part of the ecliptic, or when it comes on
the meridian at midnight, — being then about 18", — but
diminishes rapidly from the amount to about 4", which
is its apparent diameter when in conjunction, or when
seen in nearly the same direction as that luminary . This,
and facts of a similar character, observed with respect to
the apparent diameters of the other planets, clearly point
out the sun as having more than an accidental relation
to their movements.

(396.) Lastly, certain of the planets, when viewed

236 A TREATISE ON ASTRONOMY. [ciIAP. VIII.

through telescopes, exhibit the appearance of phases
like those of the moon. This proves that they are
opaque bodies, shining only by rellected light, which
can be no other than the sun's ; not only because there
is no other source of light external to them sufilciently
powerful, but because the appearance and succession of
the phases themselves are (like their visible diameters)
intimately connected with their elongations from the sun,
as will presently be shown.

(397.) Accordingly, it is found, that, when we refer
the planetary movements to the sun as a centre, all that
apparent irregularity which they offer when viewed from
the earth disappears at once, and resolves itself into one
simple and general law, of which the earth's motion, as
explained in a former chapter, is only a particular case.
In order to show how this happens, let us take the case
of a single planet, which we will suppose to revolve
round the sun, in a plane nearly, but not quite, coinci-
dent with the ecliptic, but passing through the sun, and
of course intersecting the ecliptic in a fixed line, which
is the line of the planet's nodes. This line must of
course divide its orbit into two segments ; and it is evi-
dent that, so long as the circumstances of the planet's
motion remain otherwise unchanged, tlie times of de-
scribing these segments must remain the same. The
interval, then, between the planet's quitting either node,
and returning to the same node again, must be that in
which it describes one complete revolution round the
sun, on its periodic time ; and thus we are furnished
with a direct method of ascertaining the periodic time
of each planet.

(398.) We have said (art. 388) that the planets make
the entire tour of the heavens under very different cir-
cumstances. This must be explained. Two of them —
Mercury and Venus — perform this circuit evidently as
attendants upon the sun, from whose vicinity they never
depart beyond a certain limit. They are seen sometimes
to the east, sometimes to the west of it. In the former
case they appear conspicuous over the Avestern horizon,
just after sunset, and are called evening stars : Venus,
especially, appears occasionally in this situation witli a

CHAP, nil.'] MOTIONS OF THE INFERIOR PLANETS. 237

dazzling lustre ; and in favourable circumstances may
be observed to cast a pretty strong shadow.* When
they happen to be to the west of the sun, they rise be-
fore that luminary in the morning, and appear over the
eastern horizon as morning stars : they do not, how-
ever, attain the same elongation from the sun. Mer-
cury never attains a greater angular distance from it
than aboiTt 29°, while Venus extends her excursions on
either side to about 47°. When they have receded from
the sun, eastward, to their respective distances, they
remain for a time, as it were, immoveable icith resjiect to
it, and are carried along Avith it in the ecliptic with a
motion equal to its own ; but presently they begin to
approach it, or, which comes to the same, their motion
in longitude diminishes, and the sun gains upon them.
As this approach goes on, their continuance above the
horizon after sunset becomes daily shorter, till at length
they set before the darkness has become sufficient to
allow of their being seen. For a time, then, they are
not seen at all, unless on very rare occasions, when they
are to be observed passing across the smi's disc as
small, round, loell-defined black spots, totally different
in appearance from the solar spots (art. .330). These
phenomena are emphatically called transits of the re-
spective planets across the sun, and take place when
tlie eartli happens to be passing the line of their nodes
wliile they are in that part of their orbits, just as in the
account we have given (art. 355) of a solar eclipse.
After having thus continued invisible for a time, however,
they begin to appear on the other side of the sun, at first
showing themselves only for a few minutes before sun-
rise, and gradually longer and longer as they recede from
him. At this time their motion in longitude is rapidly
retrograde. Before they attain their greatest elongation,
however, they become stationary in the heavens ; but
their recess from the sun is still maintained by the ad-
vance of that luminary along the ecliptiCi which continues
to leave them behind, until, having reversed their motion,

* It must be thrown upon a white ground. An open window if a
whitewashed room is the best exposure. In this situation, I have ob-
served not only the sliadow, but tlte dif&actcd fringes edging its outline, —

238 A TREATISE ON ASTRONOMY. [cHAP. VIII.

and become again direct, they acquire sufficient speed to
commence overtaking him— at which moment they have
their greatest ivestern elongation ; and thus is a kind of
oscillatory movement kept up, while the general advance
along the ecliptic goes on.

(399.) Suppose PQ to be the ecliptic, and ABD the
orbit of one of these planets (for instance, Mercury),
seen almost edgewise by an eye situated very nearly in
its plane ; S, the sun, its centre ; and A, B, D, S suc-
cessive positions of the planet, of which B and S are in
the nodes. If, then, the sun S stood apparently still in
the ecliptic, the planets would simply appear to oscillate
backwards and forwards from A to D, alternately passing
before and behind the sun ; and, if the eye happened to
lie exactly in the plane of the orbit, transiting his disc
in the former case, and being covered by it in the latter.
But as the sun is not so stationary, but apparently car-
ried along the ecliptic PQ, let it be supposed to move
over the spaces ST, TU, UV, while the planet in each
case executes one quarter of its period. Then will its
orbit be apparently carried along with the sun, into the
successive positions represented in the figure ; and
while its real motion round the sun brings it into the re-
spective points B, D, S, A, its apparent movement in the
heavens will seem to have been along the wavy or zig-
zag line ANHK. In this, its motion in longitude will
have been direct in the parts AN, NH, and retrograde in
the parts HnK ; while at the turns of the zigzag, at H,
K, it will have been stationary.

(400.) The only two planets — Mercury and Venus —
whose evolutions are such as above described, are called
inferior planets ; their points of farthest recess from the
sun are called (as above) their greatest eastern and west-
ern elongations; and their points of nearest approach to
it, their inferior and superior conjunctions ; the former

CHAP. VIII.] ELONGATIONS OF INFERIOR PLANETS. 230

when the planet passes between the earth and the sun,
the latter when behind the sun.

(401.) In art. 398 we have traced the apparent path
of an inferior planet, by considering its orbit in section,
or as viewed from a point in the plane of the ecliptic.
Let us now contemplate it in plan, or as viewed from a
station above that plane, and projected on it. Suppose,
then, S to represent the sun, abed the orbit of Mer-
cury, and ABCD a part of that of the earth — the direc-
tion of the circulation being the
same in both, viz. that of the
arrow. When the planet stands
at o, let the earth be situated at
A, in the direction of a tangent,
a A, to its orbit ; then it is evi-
dent that it will appear at its
greatest elongation from the
sun ; the angle oAS, M'hich
measures their apparent interval
as seen from A, being then great-
er than in any other situation of a upon its own circle.

(402.) Now, this angle being known by observation,
we are hereby furnished with a ready means of ascer-
taining, at least approximately, the distance of the planet
from the sun, or the radius of its orbit, supposed a cir-
cle. For the triansfle SArt is right-angled at a, and con-
sequently we have Sa : SA : : sin. SAa : radius, by which
proportion the radii Sa, SA of the two orbits are directly
compared. If the orbits were both exact circles, this
would of course be a perfectly rigorous mode of pro-
ceeding : but (as is proved by the inequality of the re-
sulting values of Sa obtained at different times) this is
not the case ; and it becomes necessary to admit an ec-
centricity of position, and a deviation from the exact cir-
cular form in both orbits, to account for this difference.
Neglecting, however, at present this inequality, a mean
or average value of Sa may, at least, be obtained from
the frequent repetition of this process in all varieties of
situation of the two bodies. The calculations being per-
formed, it is concluded that the mean distance of Mer-
cury from the sun is about 36000000 miles ; and that of

240 A TREATISE ON ASTRONOMY. [cHAP. VIII.

Venus, similarly derived, about G8000000 : the radius
of the earth's orbit being 95000000,

(403.) The sidereal periods of the planets may be ob-
tained (as before observed), with a considerable approach
to accuracy, by observing their passages through the
nodes of their orbits ; and, indeed, Avhen a certain very
minute motion of these nodes (similar to that of the
moon's nodes, but incomparably slower) is allowed for,
with a precision only limited by the imperfection of the
appropriate observations. By such observation, so cor-
rected, it appears that the sidereal period of Mercury is
87'* 23'^ 15'" 43-9»; and that of Venus, 224'^ 16'^ 49"
8'0'. These periods, however, are widely different from
the intervals at which the successive appearances of the
two planets at their eastern and western elongations from
the sun are observed to happen. Mercury is seen at its
greatest splendour as an evening star, at average intervals
of about 116, and Venus at intervals of about 584 days.
The difference betAveen the sidereal and synodical re-
volutions (art. 353) accounts for this. Referring again
to the figure of art. 401, if the earth stood still at A,
while the planet advanced in its orbit, the lapse of a si-
dei-eal period, which should' bring it round again to a,
would also reproduce a similar elongation from the sun.
But, meanwhile, the earth has advanced in its orbit in
the same direction towards E, and therefore the next
greatest elongation on the same side of the sun will hap-
pen— not in the position «A of the tv\'o bodies, but in
some more advanced oosition, eE. The determination
of this position depends on a calculation exactly similar
to what has been explained in the article referred to ;
and we need, therefore, only here state the resulting
synodical revolutions of the two planets, which come
out respectively 115•877^ and 583-920'^.

(404.) In this interval, the planet will have described
a whole revolution plus the arc a e, and the earth only
the arc ACE of its orbit. During its lapse, the inferior
conjunction Avill happen when the earth has a certain
intermediate situation, B and the planet has reached b, a
point between the sun and earth. The greatest elonga-
tion on the opposite side of the sun will happen when

CHAP. VIII.J SYNODICAL REVOLUTIONS. ' 241

the earth has come to C, and the planet to c, where the
line of junction Cc is a tangent to the interior circle on
tlie opposite side from M. Lastly, the superior con-
junction will happen when the earth arrives at D, and
the planet at d in the same line prolonged on the other
side of the sun. The intervals at which tliese phenome-
na happen may easily be computed from a knowledge of
the synodical periods and the radii of the orbits.

(405.) The circumferences of circles are in the propor-
tion of their radii. If, then, we calculate the circumfe-
rences of the orbits of INTercury and Venus, and the earth,
and compare them with the times in which their revolu-
tions are performed, we shall find that the actual veloci-
ties with which they move in their orbits differ greatly ;
that of Mercury being about 109400 miles per hour, of
Venus 80060, and of the earth 68080. From this it fol-
lows, that at the inferior conjunction, or at b, either
planet is moving in the same direction as the earth, but
with a greater velocity ; it will, therefore, leave the earth
behind it : and the apparent motion of the planet viewed
from the earth, will be as if tlie planet stood still, and
the earth moved in a contrary direction from what it
really does. In this situation, then, the apparent motion
of the planet must be contrary to the apparent motion of
the sun ; and, therefore, retrograde. On the other hand,
at the superior conjunction, the real motion of the planet
being in the opposite direction to that of the earth, the
relative motion will be the same as if the planet stood
still and the earth advanced with their united velocities
in its own proper direction. In this situation, then, the
apparent motion will be direct. Botli these results are in
accordance witli observed fact.

(406.) The stationary points may be determined by
the following consideration. At a or c, the points of
greatest elongation, the motion of the planet is directly
to or from the earth, or along their line of junction, while
that of the earth is nearly perpendicular to it. Here,
then, the apparent motion must be direct. At b, the in-
ferior conjunction, we have seen that it must be retro-
grade, owing to the planet's motion (which is there, as
well as the earth's, perpendicidar to the line of junction)

242 A TREATISE ON ASTRONOMY. [cHAP. Vlll.

surpassing the earth's. Hence, the stationary points
ought to lie, as it is found by observation they do, be-
tween a and b, or c and b, viz. in such a position that
tlie obliquity of the planet's motion with respect to the
line of junction shall just compensate for the excess of
its velocity, and cause an equal advance of each extre-
mity of that line, by the motion of the planet at one end,
and of the earth at the other : so that, for an instant of
time, the whole line shall move parallel to itself. The
question thus proposed is purely geometrical, and its
solution on the supposition of circular orbits is easy;
but when we regard them as otherwise than circle*^
(which they really are), it becomes somewhat complex
— too much so to be here entered upon. It will suffice
to state the results which experience verifies, and which
assigns the stationary points of Mercury at from 15° to
20° of elongation from the sun, according to circum-
stances ; and of Venus, at an elongation never varying
much from 29°. The former continues to retrograde
during about 22 days ; the latter about 42.

(407.) We have said that some of the planets exhibit
phases like the moon. This is the case with both Mer-
cury and Venus ; and is readily explained by a consi-
deration of their orbits, such as x^jg have above supposed

them. In fact, it requires little more than mere inspec-
tion of the figure annexed, to show, that to a spectator
situated on the earth E, an inferior planet, illuminated
by the sun, and therefore bright on the side next to him,
and dark on tliat turned from him, will appear /<</Z at the
superior conjunction A ; gibbous {i. e. more than half

CHAP. vm.J TRANSITS OF VENUS. 243

full, like the m8on between the first and second quarter)
between that point and the points BC of its greatest
elongation ; half-mooned at these points ; and crescent-
shaped, or horned, between these and the inferior con-
junction D. As it approaches this point, the crescent
ought to thin off till it vanishes altogether, rendering the
planet invisible, unless in those cases Avhere it transits
the sun's disc, and appears on it as a black spot. All
these phenomena are exactly conformable to observation ;
and, what is not a little satisfactory, they were predicted
as necessary consequences of the Copernican theory be«
fore the invention of the telescope.*

(408.) The variation in brightness of Venus in differ-
ent parts of its apparent orbit is very remarkable. This
arises from two causes : 1st, the varying proportion of
its visible illuminated area to its whole disc ; and, 2dly,
the varying angular diameter, or whole apparent magni-
tude of the disc itself. As it approaches its inferior con-
junction from its greater elongation, the half-moon be-
comes a crescent, which thins off; but this is more than
compensated, for some time, by the increasing apparent
magnitude, in consequence of its diminishing distance.
Thus the total light received from it goes on increasing,
till at length! it attains a maximum, wdiich takes place
when the planet's elongation is about 40°.

(409.) The fiansits of Venus are of very rare occur-
rence, taking place alternately at intervals of 8 and 113
years, or thereabouts. As astronomical phenomena, they
are, however, extremely important ; since they afford the
best and most exact means we possess of ascertaining
the sun's distance, or its parallax. Without going into
the niceties of calculation of this problem, which, owing
to the great multitude of circumstances to be attended to,
are extremely intricate, we shall here explain its prin-
ciple, which, in the abstract, is very simple and obvious.
Let E be the earth, V Venus, and S the sun, and CD the
portion of Venus's relative orbit which she describes
while in the act of transiting the sun's disc. Suppose
AB two spectators at opposite extremities of that dia-

* See Essay on the Study of Natural Puilosophy, Cab. Cyclo;
Vol. XIV. p. 269.

244 TREATISE OX ASTRONOMY. [CHAP. Vlll.

meter of the earth which is perpendicular to the ecliptic,
and, to avoid complicating the case, let us lay out of

consideration the earth's rotation, and suppose A, B, to
retain that situation during the whole time of the transit.
Then, at any moment Avhen the spectator at A sees the
centre of Venus projected at a on the sun's disc, he at B
will see it projected at b. If then one or other spectator
could suddenly transport himself from A to B, he would
see Venus suddenly displaced on the disc from a to 6 ;
and if he had any means of noting accurately the place
of the points on the disc, either by micrometrical mea-
sures from its edge, or by other means, he might ascer-
tain the angular measure of a 6 as seen from the earth.
Now, since AVa, BV6, are straight lines, and therefore
make equal angles on each side V, a b will be to AB as
the distance of Venus from the sun is to its distance from
the earth, or as G8 to 27, or nearly as 2^ to 1 : « 6, therefore,
occupies on the sun's disc a space 2^ times as great as the
earth's diameter ; and its angular measure is therefore
equal to about 2h times the earth's apparent diameter at
the distance of the sun, or (which is the same thing) to
five times the sun's horizontal parallax (art. 298). Any
error, therefore, which may be committed in measuring
a b, will entail only one ^fifth of that error on the hori-
zontal parallax concluded from it.

(410.) The thing to be ascertained, therefore, is, in
fact, neither more nor less than the breadth of the zone
PQRS, p q r s, included between the extreme apparent
paths of the centre of Venus across the sun's disc, from
its entry on one side to its quitting it on the other. The
whole business of the observers at A, B, therefore, re-
solves itself into this ; — to ascertain, with all possible
care and precision, each at his own station, this path — ■
where it enters, where it quits, and what segment of the

CHAP. Vin.J TRANSIT OF VENCS. 245

sun's disc it cuts off. Now, one of the most exact Avays
in which (conjoined with careful niicronieti-ic measures)
this can be done, is by noting the time occupied in the
whole transit : for the relative angular motion of Venus
being, in fact, very precisely known from the tables of her
motion, and the apparent path bei)ig very nearly a straight
line, these times give us a measure {on a very enlarged
scale) of the lengths of the chords of the segments cut
of!'; and the sun's diameter being known also Avith great
precision, their versed sines, and therefore their differ-
ence, or the breadth of the zone required, becomes
known. To obtain these times correctly, each observer
must ascertain the instants of ingress and egress of the
centre. To do this, he must note, 1st, the instant when
the first visible impression or notch on the edge of the
disc at P is produced, or ihe first external contact ; 2dly,
when the planet is just wholly immersed, and the
broken edge of the disc just closes again at Q, or the
first internal contact ; and lastly, he must make the same
observations at the egress at R, S. The mean of the in-
ternal and external contacts gives the entry and egress
of the planet's centre.

(411.) The modifications introduced into this process
by the earth's rotation on its axis, and by other geogra-
phical stations of the observers tliereon than here sup-
posed, are similar in their principles to those Avhich enter
into the calculation of a solar eclipse, or the occultation of
a star by the moon, only more refined. Any considera-
tion of them, however, here, would lead us too far ; but
in the view we have taken of the subject, it affords an
admirable example of the way in which minute elements
in astronomy may become magnified in their efl^ects, and,
by being made subject to measurement on a greatly en-
larged scale, or by substituting the measure of time for
space, may be ascertained with a degree of precision
adequate to every purpose, by only watching favourable
opportunities, and taking advantage of nicely adjusted
combinations of circumstance. So important has this
observation appeared to astronomers, that at the last
transit of Venus, in 1769, expeditions were fitted out, on
the most efficient scale, by the British, French, Russian,

x2

246 A TKEATI6E ON ASTRONOMY. [cHAP. VIII.

and other governments, to the rcmotost corners of the
globe, for the express purpose of performing it. The
celebrated expedition of Captain Cook to Otaheite was
one of them. The general result of all the observations
made on this most memorable occasion gives. 8"-5776
for the sun's horizontal parallax.

(412.) The orbit of Mercury is very elliptical, the ec-
centricity being nearly one fourdi of the mean distance.
This appears from the inequality of the greatest elonga-
tions from the sun, as observed at difterent times, and
which vary between the limits 16° 12' and 28° 48', and,
from exact measures of such elongations, it is not diffi-
cult to show that the orbit of Venus also is slightly ec-
centric, and that both these planets, in fact, describe
ellipses, having the sun in their common focus.

(413.) Let us now consider the superior planets, or
those whose orbits enclose on all sides that of the earth.
That they do so is proved by several circumstances : —
1st, They are not, like the inferior planets, confined to
certain limits of elongation from the sun, but appear at
all distances from it, even in the opposite quarter of tlie
heavens, or, as it is called, in opposition ; which could
not happen, did not the earth at such times place itself
between them and the sun : 2dly, They never appear
horned, like Venus or Mercury, nor even semUunar.
Those, on the contrary, which, from the minuteness of
their parallax, we conclude to be the most distant from
us, viz. Jupiter, Saturn, and Uranus, never appear other-
wise than round ; a sufficient proof, of itself, that we see
them always in a direction not very remote from that in
Avhich the sun's rays illuminate them ; and that, there-
fore, we occupy a station which is never very widely re-
moved from the centre of their orbits, or, in other words,
that the earth's orbit is entirely enclosed within theirs,
and of comparatively small diameter. One only of them,
Mars, exhibits any perceptible phase, and in its defi-
ciency from a circular outline, never surpasses a mode-
rately gibbous appearance — the enlightened portion of
the disc being never less than seven-eighths of the whole.
To understand this, we need only cast our eyes on the
annexed figure, in which E is the earth, at its apparent

CHAP. VIII.] DISTANCES OF SUPERIOR PLANETS. 347

greatest elongation from the sun S, as
seen from Mars, M. In this position,
the angle SME, included between tlie
lines SM and EM, is at its maximum;
and, therefore, in this state of things, a
spectator on the earth is enabled to see a
greater portion of the dark hemisphere
of Mars than in any other situation. The
extent of the phase, then, or greatest ob-
servable degree of gibbosity affords a
measure — a sure, although a coarse and
rude one — of the angle SME, and there-
fore of the proportion of the distance
SM, of Mars to SE, that of the earth
from the sun, by which it appears that
the diameter of the orbit of Mars can- M

not be less than I2 that of the earth's. The phases of
Jupiter, Saturn, and Uranus being imperceptible, it fol-
lows that their orbits must include not only that of the
earth, but of Mars also.

(414.) All the superior planets are retrograde in their
apparent motions when in opposition, and for some time
before and after ; but they differ greatly from each other,
both in the extent of their arc of retrogradation, in the
duration of their retrograde movement, and in its rapidity
when swiftest. It is more extensive and rapid in the
case of Mars than of .Jupiter, of Jupiter than of Saturn,
and of that planet than Uranus. The angtdar velocity
with which a planet appears to retrograde is easily ascer-
tained by observing its apparent place in the heavens
from day to day ; and from such observations, made about
the time of opposition, it is easy to conclude the relative
magnitudes of their orbits as compared with the earth's,
supposing their periodic times known. For, from these,
their mean angular velocities are known also, being in-

versely as the times. Suppose, then, Ee to be a very
small portion of the earth's orbit, and Mm a correspond-

248 A TREATISE ON ASTRONOMY. [cHAP. VIII,

ing portion of that of a superior planet, described on the
day of opposition, about t!ie sini S, on which day the
three bodies lie in one straight line SEMX. Then the
angles ESe and MSm ai-e given. Now, if e m be joined
and prolonged to meet SM continued in X, the angle eXE,
Avhich is equal to the alternate angle Xe?/, is evidently the
retrogradation of Mars on that day, and is, therefore, also
given. Ee, therefore, and the angle EXe, being given in
the right-angled triangle EeX, the side EX is easily cal-
culated, and thus SX becomes known. Consequently,
in the triangle SmX, we have given the side SX and the
two angles j»SX and wXS, whence the other sides, S?w,
mK, are easily determined. Now, S?7i is no other than
the radius of the orbit of the superior planet required,
which in this calculation is supposed circular as well as
that of the earth ; a supposition not exact, but sufficiently
so to afford a satisfactory approximation to the dimen-
sions of its orbit, and which, if the process be often re-
peated, in every variety of situation at which the oppo-
sition can occur, will ultimately afford an average or
mean value of its diameter fully to be depended upon.

(415.) To apply this principle, however, to practice,
it is necessary to know the periodic times of the several
planets. These may be obtained directly, as has been
already stated, by observing the intervals of their pas-
sages through the ecliptic ; but owing to the very small
inclination of the orbits of some of them to its plane,
they cross it so obliquely that the precise moment of
their arrival on it is not ascertainable, unless by very nice
observations. A better method consists in determining,
from the observations of several successive days, the
exact moments of their arriving in opj)Osition with the sun,
the criterion of which is a difference of longitudes be-
tween the sun and planet of exactly 180°. The interval
between successive oppositions thus obtained is nearly
one synodical period ; and would be exactly so, were the
planet's orbit and that of the earth both circles, and uni-
formly described ; but as that is found not to ,be the case
(and the criterion is", the ineqiicdity of successive synod-
ical revolutions so observed), tlie average of a great num-
ber, taken in all varieties of situation in which the oppo«

CHAP. viii.J Kepler's law of periodic times. 349

sitions occur, will be freed from tlie elliptic inequality,
and may be taken as a mean synocUcal period. From
this, by the considerations employed in art. 353, and by
the process of calculation indicated in the note to that
article, the sidereal periods are readily obtained. The
accuracy of this determination will, of course, be greatly
increased by embracing a long interval between the ex-
treme observations employed. In point of fact, that in-
terval extends to nearly 2000 years in the cases of the
planets known to the ancients, who have recorded their
observations of them in a manner sufficienUy careful to
be made use of. Their periods may, therefore, be regard-
ed as ascertained w^ith the utmost exactness. Their nu-
merical values will be found stated, as well as the mean
distances, and all the other elements of the planetary
orbits, in the synoptic table at the end of the volume, to
which (to avoid repetition) the reader is once for all re-
ferred.

(416.) In casting our eyes down the list of the planet-
ary distances, and comparing them with the periodic
times, we cannot but be struck with a certain correspond-
ence. The greater the distance, or the larger the orbit,
evidently the longer the period. The order of the pla-
nets, beginning from the sun, is the same, whether we
arrange them according to their distances, or to the time
they occupy in completing their revolutions ; and is as
follows : — Mercury, Venus, Earth, Mars — the four ultra-
zodiacal planets — Jupiter, Saturn, and Uranus. Never-
theless, when we come to examine the numbers express-
ing them, we find that the relation between the two series
is not that of simple proportional increase. The periods
increase more than in proportion to the distances. Thus,
the period of Mercury is about 88 days, and that of the
Earth 365 — being in proportion as 1 to 4-15, while their
distances are in the less proportion of 1 to 3-56 ; and a
similar remark holds good in every instance. Still, the
ratio of increase of the times is not so rapid as that of
the squares of the distances. The square of 2-56 is
6'5536, which is considerably greater than 4"15. An in-
termediate rate of increase, between the simple proportion
of the distances and that of their squares, is therefore

250 A TREATISE ON ASTRONOMY. |^CHAF. VIII.

clearly pointed out by the sequence of the numbers ; but
it required no ordinary penetration in the illustrious Kep-
ler, backed by uncommon perseverance and industry, at
a period wlien the data themselves were involved in ob-
scurity, and when the processes of trigonometry and of
numerical calculation were encumbered with difficulties,
of which the more recent invention of logarithmic tables
has happily left us no conception, to perceive and demon-
strate the real law of their connexion. This connexion
is expressed in the following proposition : — " The squares
of the periodic times of any two planets are to each
other, in the same proportion as the cubes of their mean
distances from the sun." Take, for example, the earth
and Mars,* v/hose periods are in the proportion of
3652564 to 6869796, and whose distances from the sun
is that of 100000 to 152369 ; and it will be found, by
any one who will take the trouble to go through the calcu-
lation, that —

(3652564)2: (6869796)-:: (100000) 3; (152369)3.
(417.) Of all the laws to which induction from pure
observation has ever conducted man, this third law (as
it is called) of Kepler may justly be regarded as the most
remarkable, and the most pregnant with important conse-
quences. When we contemplate the constituents of the
planetary system from the point of view which tliis rela-
tion affords us, it is no longer mere analogy Avhich strikes
us — no longer a general resemblance among them, as
individuals independent of each other, and circulating
about the sun, each according to its own peculiar nature,
and connected with it by its own peculiar tie. Tire re-
semblance is iioAV perceived to be a true family likeness ;
they are bound up in one chain — -interwoven in one web of
mutual relation and harmonious agreement— subjected to
one pervading influence, which extends from the centre
to the farthest limits of that great system, of which all of
them, the earth included, must henceforth be regarded as

members.

(418.) The laws of elliptic motion about the sun as a

* The expression of this law of Kepler requires a slight modificalion

when we come to the extreme nicety of numerical calculation, for the

greater iilanets, due to the inOuence of their masses. This correction i^

<j!Qperceplible for the earth oxid Mars.

CHAP. Vin.;] INTERPRETATION OP KEPLER's LAWS. 251

focus, and of the equable description of areas by lines
joining the sun and planets, were originally established
by Kepler, from a consideration of the observed motions
of Mars ; and were by him extended, analogically, to all
the other planets. However precarious such an extension
might then have appeared, modern astronomy has com-
pletely verified it as a matter of fact, by the general coinci-
dence of its results with entire series of observations of
the apparent places of the planets. These are found to
accord satisfactorily with the assumption of a particular
elUpse for each planet, whose magnitude, degree of eccen-
tricity, and situation in space, are numerically assigned
in the synoptic table before referred to. It is true, that
when observations are carried to a high degree of preci-
sion, and when each planet is traced through many suc-
cessive revolutions, and its history carried back, hy the
aid of calculations founded on these data, for many centu-
ries, we learn to regard the laws of Kepler as only Jirst
approximations to the much more complicated ones
which actually prevail ; and that to bring remote observa-
tions into rigorous and mathematical accordance with
each other, and at the same time to retain the extremely
convenient nomenclature and relations of the elliptic
SYSTEM, it becomes necessary to modify, to a certain ex-
tent, our verbal expression of the laws, and to regard the
numerical data or elliptic elements of the planetary orbits
as not absolutely permanent, but subject to a series of
extremely slow and almost imperceptible changes. These
changes may be neglected when we consider only a few
revolutions ; but going on from century to century, and
continually accumulating, they at length produce consider-
able departures in the orbits from their original state.
Their explanation will form the subject of a subsequent
chapter ; but for the present we must lay them out of
consideration, as of an order too minute to affect the gene-
ral conclusions with which we are now concerned. By
what means astronomers are enabled to compare the re-
sults of the elliptic theory with observation, and thus
satisfy themselves of its accordance with nature, will be
explained presently.

(419.) It will first, however, be proper to point out

252 A TREATISE ON ASTRONOMY. [[cHAP. Vlll.

what particular theoretical conclusion is involved in each
of the three laws of Kepler, considered as satisfactorily
established, — what indication each of them separately
affords of the mechanical forces prevalent in our system,
and the mode in Avhich its parts are connected — and how,
when thus considered, they constitute the basis on which
the Newtonian explanation of the mechanism of the hea-
vens is mainly supported. To begin with the first law,
that of the equable description of areas, — Since the pla-
nets move in curvilinear paths, thej must (if they be bo-
dies obeying the laws of dynamics) be deflected from
their otherwise natural rectilinear progress by force. And
from this law, taken as a matter of observed fact, it fol-
lows, that the direction of such force, at every point of
the orbit of each planet, always passes through the sun.
No matter from what ultimate cause the poAver which is
called gravitation originates — be it a virtue lodged in
the sun as its receptacle, or be it pressure from without,
or the resultant of many pressures or solicitations of un-
known fluids, magnetic or electric ethers, or impulses —
still, when finally brought under our contemplation, and
summed up into a single resultant energy, its direction
is, from every point on all sides, towards the sicn''s cen-
tre. As an abstract dynamical proposition, the reader
■will find it demonstrated by Newton, in the 1st proposi-
tion of the Principia, with an elementary simplicity to
which we really could add nothing but obscurity by ampli-
fication, that any body, urged towards a certain central
point by a force continually directed thereto, and thereby
deflected into a curvilinear path, will describe about that
centre equal areas in equal times ; midi vice versa, that
such equable description of areas is itself the essential
criterion of a continual direction of tlie acting force to-
wards the centre to wliich this character belongs. The
first law of Kepler, then, gives us no information as to the
nature or intensity of the force urging the planets to the
sun ; the only conclusion it involves, is that it does so
urge them. It is a property of orbitual rotation under
the influence of central forces generally, and as such, we
daily see it exemplified in a thousand familiar instances.
A simple experimental illustration of it is to tie a bullet

CHAl'. VIII.] INTERPREtAtiON OF KEPLEr's LAWS. 253

to a thin string, and, having whirled it round with a mo-
derate velocity in a vertical plane, to draw the end of the
string through a small ring, or allow it to coil itself round
the finger, or a cylindrical rod held very firmly in a hori-
zontal position. The bullet will then approach the centre
of motion in a spiral line ; and the increase not only of its
angular but of its linear velocity, and the rapid diminution
of its periodic time when near the centre, will express,
more clearly than any words, the compensation by which
its uniform description of areas is maintained under a
constantly diminishing distance. If the motion be re-
versed, and the thread allowed to uncoil, beginning with
a rapid impulse, the velocity will diminish by the same
degrees as it before increased. The increasing rapidity
of a dancer's jnroucttc>, as he draws in his limbs and
straightens his whole person, so as to bring every part of
his frame as near as possible to the axis of his motion, is
another instance where the connexion of the observed
effect with the central force exerted, though equally real,
is much less obvious.

(420.) The second law of Kepler, or that which as-
serts that the planets describe ellipses about the sun as
their focus, involves, as a consequence, the law of solar
gravitation (so be it allowed to call the force, whatever it
be, which urges them towards the sun) as exerted on each
individual planet, apart from all connexion with the rest.
A straight line, dynamically speaking, is the only path
which can be pursued by a body absolutely free, and un-'
der the action of no external force. All deflection into a
curve is evidence of the exertion of a force ; and the
greater the deflection in equal times, the more intense the
force. Deflection from a straight line is only another
word for curvature of path ; and as a circle is character^'
ized by the uniformity of its curvature in all its parts — so
is every other curve (as an ellipse) characterized by the
particular law which regulates the increase and diminu''
tion of its curvature as we advance along its circumfe^
rence. The deflecting force, then, which continually
bends a moving body into a curve, may be ascertained,
provided its direction, in the first place, and, secondly,
the law of curvature of the curve itself, be known. Both
these enter as elements into the expression of the force. A

Y

254 A TREATISE ON ASTRONOMY. [cHAP. Mil.

body may describe, for instance, an ellipse, under a great
variety of dispositions of the acting forces : it may glide
along it, for example, as a bead upon a polished wire,
bent into an elliptic form ; in which case the acting force
is always perpendicular to the wire, and the velocity is
uniform. In this case the force is directed to 7io fixed
centre, and there is no equable description of areas at all.
Or it may describe it as we may see it done, if we sus-
pend a ball by a very long string, and, drawing it a little
aside from the perpendicular, tlirow it round with a gen-
tle impulse. In this case the acting force is directed to
the centre of the ellipse, about which areas are described
equably, and to which a force jjroportional to the distance
(the decomposed result of terrestrial gravity) perpetually
urges it. This is at once a very easy experiment, and a very
instructive one, and we shall again refer to it. In the
case before us, of an ellipse described by the action of a
force directed to the focus, the steps of the investigation
of the law of force are these : 1st, The law of the areas
determines the actual velocity of the revolving body at
every point, or the space really run over by it in a given
minute portion of time ; 2dly , The law of curvature of the
ellipse determines the linear amount of deflection from the
tangent in the direction of the focus, which corresponds
to that space so run over ; .3dly, and lastly. The laws of
accelerated motion declare that the intensity of the acting
force causing such deflection in its own direction, is mea-
sured by or proportional to the amount of that deflection,
and may therefore be calculated in any particular position,
or generally expressed by geometrical or algebraic sym-
bols, as a law independent of particular positions, when
that deflection is so calculated or expressed. We have
here the spirit of the process by which Newton has resolved
this interesting problem. For its geometrical detail, we
must refer to the 3d section of his Principia. We know
of no artificial mode of imitating this species of elliptic
motion ; though a rude approximation to it — enough,
however, to give a conception of the alternate approach
and recess of the revolving body to and from the focus,
and the variation of its velocity — may be had by suspend-
ing a small steel bead to a fine and very long silk fibre,
and setting it to revolve in a small orbit round the pole of

CHAP. Vin.] INTERPRETATION OF KEPLEr's LAWS. 255

a powerful cjlindrical magnet, held upright, and verti-
cally under the point of suspension.

(421.) The third law of Kepler, which connects the
distances and periods of the planets by a general rule,
bears with it, as its theoretical interpretation, this im-
portant consequence, viz. that it is one and the same
force, modified only by distance from the sun, which
retains all the planets in their orbits about it. That the
attraction of the sun (if such it be) is exerted upon all
the bodies of our system indifferently, without regard to
the peculiar materials of which they may consist, in the
exact proportion of their inertise, or quantities of matter;
that it is not, therefore, of the nature of the elective at-
tractions of chymistry, or of magnetic action, which is
powerless on other substances than iron and some one
or two more, but is of a more universal character, and
extends equally to all the material constituents of our
system, and (as we shall hereafter see abundant reason to
admit) to those of other systems than our own. This
law, important and general as it is, results, as the sim-
plest of corollaries, from the relations established by
Newton in the section of the Principia referred to
(prop. XV.), from which proposition it results, that if
the earth were taken from its actual orbit, and launched
anew in space at the place, in the direction, and with
the velocity of any of tlie other planets, it would describe
the. very same orbit, and in the same period, which that
planet actually does, a very minute correction of the pe-
riod only excepted, arising from the diflerence between
the mass of the earth and that of the planet. Small as the
planets are compared to the sun, some of them are not,
as the earth is, mere atoms in the comparison. The
strict wording of Kepler's law, as Newton has proved in
his fifty-ninth proposition, is applicable only to the case
of planets whose proportion to the central body is abso-
lutely inappreciable. When this is not the case, the
periodic time is shortened in the proportion of the
square root of the number expressing the sun's mass
or inertia, to that of the sum of the numbers expressing
the masses of the sun and planet; and in general, what-
ever be the masses of two bodies revolving round each
other under the influence of the Newtonian law of gra-

256 A TREATISE ON ASTRONOMY. [cHAP. VIII.

vity, the square of their periodic time will be expressed
by a fraction whose numerator is tlie cube of their mean
distance, i. c. the greater semi-axis of their elliptic orbit,
and whose denominator is the sum of their masses.
When one of the masses is incojnparably greater than
the other, this resolves itself into Kepler's law ; but
when this is not the case, the proposition thus general-
ized stands in lieu of that law. In the system of the sun
find planets, however, the numerical correction thus in
troduced into the results of Kepler's law is too small to
be of any importance, the mass of the largest of the
planets (Jupiter) being much less than a thousandth
part of that of the sun. We shall presently, however,
perceive all the importance of this generalization, when
\ye come to speak of the satellites.

(422.) It will first, ]iov>^cver, be proper to explain by
what process of calculation the expression of a planet's
elliptic orbit by its elements can be compared with ob-
servation, and hoAV we can satisfy ourselves that the
jiumerical data contained in a table of such elements for
the whole system does really exhibit a tnie picture of
Jt, and afford the means of determining its state at every
Instant of time, by the mere applieation of Kepler's laws.
Now, for each planet, it is necessary for this purpose to
]{now, 1st, the magnitude and form of its ellipse ; 2dly,
ihe situation of this ellipse in space, with respect to the
ecliptic, and to a fixed line drawn therein ; 3dly, the
Jocal situation of the planet in its ellipse at some known
ppoch, and its periodic time or mean angular velocity,
pr, as it is called, its mean motion.

(423.) The magnitude and form of an ellipse are de-.
termined by its greatest length and least breadth, or its
two principal axes ; but for astronomical uses it is pre-!
ferable to use the semi-axis major (or half the greatest
length), and the eccentricity or distance of the focus,
from the centre, which last is usually estimated in parts
of the former. Thus, an ellipse, whose length is 10
and breadth 8 parts of any scale, has for its major semi-,
axis 5, and for its eccentricity 3 such parts ; but when
estimated in parts of the semi-axis, regarded as a unit,
the eccentricity is expressed by the fraction |.

^434)' The ecliptic is the plane tp which ?in ii^liabit-s

CHAP. VIII.] ELEMENTS OF A PLANEx's ORBIT. 257

ant of the earth most naturally refers the rest of the solar
system, as a sort of ground-plane ; and the axis of its
orbit might be taken for a line of departure in that plane
or origin of angular reckoning. Were the axis Jixed,
this would be the best possible origin of longitudes ; but
as it has a motion (though an excessively slow one),
there is, in fact, no advantage in reckoning from the axis
more than from the line of the equinoxes, and astrono-
mers therefore prefer the latter, taking account of its va-
riation by the effect of precession, and restoring it, by
calculation at every instant, to a fixed position. Now,
to determine the situation of the ellipse described by a
planet with respect to this plane, three elements require
to be known: — 1st, the indbtation of the plane of the
planet's orbit to the plane of the ecliptic ; 2dly, the line
in which these two planes intersect each other, which of
necessity passes through the sun, and whose position
with respect to the line of the equinoxes is therefore
given l)y stating its longitude. This line is called the
Une of the nodes. When the planet is in tliis line, in
the act of passing from the south to the north side of
the ecliptic, it is in its ascending node, and its longitude
at that moment is the element called the longitude of the
node. These two data determine the situation of t/ie
plane of the orbit ; and there only remains, for the com-
plete determination of the situation of the planet's ellipse,
to know how it is placed in that plane, which (since its
focus is necessarily in the sun) is ascertained by stating
the longitude of its perihelion, or the place which the
extremity of the axis nearest the sun occupies, when
orthographically projected on the ecliptic.

(425.) The dimensions and situation of the planet's
orbit thus determined, it only remains, for a complete
acquaintance with its history, to determine the circum-
-.. stances of its motion in the orbit so precisely fixed.
Now, for this purpose, all that is needed is to know the
moment of time when it is either at the perihelion, or
at any other precisely determined point of its orbit, and
its whole period ; for these being known, the law of the
areas determines the place at every other instant. This
moment is called (when the perihelion is the point
chosen) the perihelion passage, or, when some point of

258 A TUEATISE ON ASTRONOMY. [ciIAP. VIIl.

the orbit is fixed upon, without special reference to the
perihelion, the epoch.

(426.) Thus, then, we have seven particulars or ele-
ments, which must be numerically stated, before we can
reduce to calculation the state of the system at any
given moment. But, these known, it is easy to ascertain
the apparent positions of each planet, as it wovild be seen
from the sun, or is seen from the earth at any time.
The former is called the heliofentric, the latter the geo-
centric, place of the planet.

(427.) To commence with the
heliocentric places. Let S re-
ly present the sun ; APN the orbit
i-^/'of the planet, being an ellipse,
having the sun S in its focus,
and A for its perihelion ; and let
]f)aN V represent the projection of the orbit on the plane
of the ecliptic, intersecting the line of equinoxes S V in
T» which, therefore, is the origin of longitudes. Then
will SN be the line of nodes ; and if we suppose B to
lie on the south, and A on the north side of the ecliptic,
and the direction of the planet's motion to be from B to
A, N will be the ascending node, and the angle T SN the
iongitude of the node. In like manner, if P be the place
of the planet at any time, and if it and the perihelion A
be projected on the ecliptic, upon the points yj a, the angles
*\p 8p, T Sa, will be the respective heliocentric longitudes
of the planet, and of the perihelion, the former of whicli
is to be determined, and the latter is one of the given
elements. Lastly, the angle ;jSP is the heliocentric lati-
tude of the planet, which is also required to be known.

(428.) Now, the time being given, and also the mo-
ment of the planet's passing the perihelion, the interval,
or the time of describing the portion AP of the orbit, is
given, and the periodical time, and the whole area of the
ellipse being known, the law of proportionality of areas
to the times of their description gives the magnitude of
the area ASP, From tliis it is a problem of pure geo-
metry to determine the corresponding angle ASP, which
is called the planet's true anomaly. This problem is of
the kind called transcendental, and has been resolved by
a great variety of processes, some more, some less in-

CHAP. Vin.3 HELIOCENTRIC PLACE OF A PLANET. 259

tricate. It ofiers, however, no peculiar difRcnlty, and is
practically resolved with great facility by the help of
tables constructed for the purpose, adapted to the case of
each particular planet.*

(429.) The true anomaly thus obtained, the planet's
angular distance from the node, or the angle NSP, is to
be found. Now, the longitudes of the perihelion and
node being respectively T a and T N, which are given,
their difl'trence «N is also given, and the angle N of the
spherical right-angled triangle AN«, being the inclina-
tion of the plane of the orbit to the ecliptic, is known.
Hence we calculate the arc NA, or the angle NSA,
wiiich, added to ASP, gives the angle NSP required.
And from this, regarded as the measure of the arc NP,
forming the hypothenuse of the right-angled spherical
triangle PN/j, whose angle N, as before, is known, it
is easy to obtain the other two sides, N/3 and Pp. The
latter, being the measure of the angle ^^SP, expresses
the planet's heliocentric latitude ; the former measures
the angle NS;;, or the planet's distance in longitude
from its node, which, added to the known angle T SN,
the longitude of the node, gives the heliocentric longitude.
This process, however circuitous it may appear, when
once well understood, may be gone through numerically,
by the aid of the usual logarithmic and trigonometrical
tables, in little more time than it will have taken the
reader to peruse its description.

(4.30.) The geocentric differs from the heliocentric

place of a planet by reason of that parallactic change of

apparent situation which arises from the earth's motion

in its orbit. Were the planets' distance as vast as those

* It will readily be understood, that, except in the case of uniform cir-
cular motion, an equable description of areas about any centre is incom-
patible with an equable description of o«^fes. The object of the problem
in the text is to pass from the area, supix>sed knovvii, to the angle, sup-
posed unknown : in other words, to derive the true amoimt of angular
motion from the perihelion, or the true anomalii from what is teehnically
called the mean anomaly, that is, the mean angular motion which would
have been performed liad the motion in angle been unifonn instead of
the motion in area. It happens, fortunately, that this is the simplest of
all problems of the transcendental kind, and can be resolved, in the
most diflicult case, by the rule of" fiilse position," or trial and error, in a
very few minutes. Nay, it may even be resolved instantly on inspec-
tion by a simple and easily constructed piece of mechanism, of which the
reader may see a descrijition in tlie Cambridge Philosophical Transao
tions, vol- IV. p. 425, by tbe author of this work.

260 A TREATISE ON ASTRONOMY. [ciIAP. VIII.

of the stars, the earth's orbitual motion would be insen-
sible when viewed from them, and they would always
appear to us hold the same relative situations among the
fixed stars, as if viewed from the sun, i. e. they would
then be seen in their heliocentric places. The differ-
ence, then, between the heliocentric and geocentric
places of a planet is, in fact, the same thing with its pa-
rallax arising from the earth's removal from the centre
of the system and its annual motion. It follows from
this, that the first step towards a knowledge of its
amount, and the consequent determination of the ap-
parent place of each planet, as referred from the earth to
the sphere of the fixed stars, must be to ascertain the
proportion of its linear distances from the earth and
from the sun, as compared with the earth's distance from
the sun, and the angular positions of all three with re-
spect to each other.

(431.) Suppose, therefore, S to represent the sun, E
the earth, and P the planet ; S T the line of equinoxes,
T E the earth's orbit, and Pp a perpendicular let fall
from the planet on the ecliptic. Then will the angle
SPE (according to the general notion of parallax con-
veyed in art. 69) represent the parallax of the planet

arising from the change of sta-
tion from S to E, EP will be
the apparent direction of the
Jfi planet seen from E ; and if SQ
be drawn parallel to E/>, the
angle T SQ will be the geo-
centric longitude of the planet,
while T SE represents the heliocentric longitude of the
earth, and T ^p that of the planet. The former of
these, T SE, is given by the solar tables ; the latter,
T Sp is found by the process above described (art. 429).
Moreover, SP is the radius vector of the planet's orbit,
and SE that of the earth's, both of whicli are determined
from the known dimensions of their respective ellipses,
and the places of the bodies in them at the assigned time.
Lastly, the angle VSp is the planet's heliocentric lati-
tude.

(432.) Our object, then, is, from all these data, to de-
termine the anffle T SQ and PEp, which is the geocen-

CHAP. VIII,] DISCOVERY OF THE PLANETS. 261

trie latitude. The process, then, will stand as follows :
1st, In the triiinixle SP/j, right-ang-led at P, given SP,
and the angle PS/; (the planet's radius vector and helio-
centric latitude,) find Sp, and Pp ; 2dly, In the triangle
SE;;, given S/j (just found), SE (the earth's radius
vector), and the angle ES/J (the difference of heliocen-
tric longitudes of the earth and planet), find the angle
S/)E, and the side E7J. The former being eqival to the
alternate angle />SQ, is the parallactic removal of the
planet in longitude, which, added to T Sp, gives its helio"
centric longitude. The latter, E^; (which is called the
curtate distance of the planet from the earth), gives at
once the geocentric latitude, by means of the right-angled
triangle FEp, of which Ep and Fp are known sides,
and the angle FEp is the longitude sought.

(433.) The calculations required for these purposes
are nothing but the most ordinary processes of plane
trigonometry ; and, though somewhat tedious, are nei-
ther intricate nor difficult. When executed, however,
they afford us the means of comparing the places of
the planets actually observed with the elliptic theory,
with the utmost exactness, and thus putting it to the se-
verest trial ; and it is upon the testimony of such compu-
tations, so brought into comparison with observed facts,
that we declare that theory to be a true representation of
nature.

(434.) The planets Mercury, Venus, Mars, Jupiter,
and Saturn, have been known from the earliest ages in
which astronomy has been cultivated. Uranus Avas dis-
covered by Sir W. Herschel in 1781, March 13, in the
course of a review of the heavens, in which every star
visible in a telescope of a certain power was brought
under close examination, when the new planet was im-
mediately detected by its disc, under a high magnifying
power. It has since been ascertained to have been ob-
served on many previous occasions, with telescopes of
insufficient power to show its disc, and even entered in
catalogues as a star ; and some of the observations which
have been so recorded have been used to improve and
extend our knowledge of its orbit. The discovery of the
ultra-zodiacal planets dates from the first day of 1801,
when Ceres was discovered by Piazzi, at Palermo ; g

262 A TREATISE ON ASTRONOMY. [cHAP. VIII.

discovery speedily followed by those of Juno by Pro-
fessor Harding, of Gottingen ; and of Pallas and Vesta,
by Dr. Olbers, of Bremen. It is extremely remarkable
that this important addition to our system had been in
some sort surmised as a thing not unlikely, on the ground
that the intervals between the planetary orbits go on
doubling as we recede from the sun, or nearly so. Thus,
the interval between the orbits of the earth and Venus is
nearly twice that between those of Venus and Mercury ;
that between the orbits of Mars and the earth nearly
twice that between the earth and Venus ; and so on.
The interval between the orbits of Jupiter and Mars,
however, is too great, and would form an exception to
this law, which is, however, again resumed in the case
of the three remoter planets. It was, therefore, thrown
out, by the late Professor Bode of Berlin, as a possible
surmise, that a planet might exist between Mars and
Jupiter ; and it may easily be imagined what was the as-
tonishment of astronomers to find four, revolving in orbits
tolerably well corresponding with the law in question.
No account, ci priori, or from theory, can be given of this
singular progression, which is not, like Kepler's laws,
strictly exact in its numerical verification ; but the cir-
cumstances we have just mentioned lead to a strong be-
lief that it is something beyond a mere accidental coinci-
dence, and belongs to the essential structure of the
system. It has been conjectured that the ultra-zodiacal
planets are fragments of some greater planet, which
formerly circulated in that interval, but has been blown
to atoms by an explosion ; and that more such fragments
exist, and may be hereafter discovered. This may
serve as a specimen of the dreams in which astronomers,
like other speculators, ccasionally and harmlessly indulge.
(435.) We shall devote the rest of this chapter to an
account of the physical peculiarities and probable condi-
tion of the several planets, so far as the former are known
by observation, or the latter rest on probable grounds of
conjecture. In this, three features principally strike us,
as necessarily productive of extraordinary diversity in the
provisions by which, if they be, like our earth, inhabited,
animal life must be supported. There are, first, the dif-
ference in their respective supplies of light and heat from

CHAP. VIII.3 APPEARANCES OF THE PLANETS. 263

the sun ; secondly, the difference in the intensities of the
gravitating forces which must subsist at their surfaces, or
the different ratios which, on their several globes, the
inertiae of bodies must bear to their weights ; and, third-
ly, the difference in the nature of the materials of which,
from what we know of their mean density, we have
every reason to believe they consist. The intensity of
solar radiation is nearly seven times gi-eater on Mercury
than on the eartli, and on Uranus 330 times less ; the
proportion between the two extremes being that of
upwards of 2000 to one. Let any one figure to himself
the condition of our globe, were the sun to be septupled,
to say nothing of the greater ratio ! or were it diminished
to a seventh, or to a 300th of its actual power ! Again,
the intensity of gravity, or its efficacy in counteracting
muscular power and repressing animal activity on Jupiter
is nearly three times that on the Earth, on Mars not more
than one third, on the Moon one sixth, and on the four
smaller planets probably not more than one twentieth ;
giving a scale of which the extremes are in the proportion
of sixty to one. Lastly, the density of Saturn hardly
exceeds one eighth of the mean density of the earth, so
that it must consist of materials not much heavier than
cork. Now, under the various combinations of elements
so important to life as these, what immense diversity
must we not admit in the conditions of that great problem,
the maintenance of animal and intellectual existence and
happiness, which seems, so far as we can judge by what
we see around us in our own planet, and by the way in
which every corner of it is crowded with living beings, to
form an unceasing and worthy object for the exercise of
the Benevolence and Wisdom which presides over all !

(436.) Quitting, however, the region of mere specula-
tion, we will now show what information the telescope
affords us of the actual condition of the several planets
within its reach. Of Mercury we can see little more than
that it is round, and exhibits phases. It is too small,
and too much lost in the constant neighbourhood of the
Sun, to allow us to make out more of its nature. The
real diameter of Mercury is about 3200 miles : its appa-
rent diameter varies from 5" to 12". Nor does Venus
offer any remarkable peculiarities : although its real dia-

264 A TREAflSE ON ABTliOKOMY. [cHAt'. Vltl.

meter is 7800 miles, and although it occasionally attains
the considerable apparent diameter of 61", which is
larger than that of any other planet, it is yet the most dif-
ficult of them all to define with telescopes. The intense
lustre of its illuminated part dazzles the sight, and exag*
gerates every imperfection of the telescope ; yet we see
clearly that its surface is not mottled over with permanent
spots like the moon ; Ave perceive in it neither mountains
nor shadows, but a uniform brightness, in which some-
times we may, indeed, fancy obscurer portions, but can
seldom or never rest fully satisfied of the fact. It is from^
some observations of this kind that both Venus and Mer^
cury have been concluded to revolve on their axes in
about the same time as the Earth. The most natural
conclusion, from the very rare appearance and want of
permanence in the spots, is, that we do not see, as in the
Moon, the real surface of these planets, but only their
atmospheres, much loaded with clouds, and which may
serve to mitigate the otherwise intense glare of their sun-
shine.

(437.) The case is very different with Mars. In this
planet we discern, Avith perfect distinctness, the outlines
of Avhat may be continents and seas. (See plate I. fig,
1, Avhich represents Mars in its gibbous state, as seen on
the 16th of August, 1830, in the 20-feet reflector at Slough.)
Of these, the former are distinguised by that ruddy colour
which characterizes the light of this planet (which ahvays
appears red and fiery), and indicates, no doubt, an ochrey
tinge in the general soil, like Avhat the red sandstone dis-
tricts on the Earth may possibly offer to the inhabitants
of Mars, only more decided. Contrasted Avitli this (by
a general law in optics), the seas, as we may call them,
appear greenish.* These spots, hoAvever, are not ahvays
to be seen equally distinct, though, lohen seen, they offer
ahvays the same appearance. This may arise from the pla-
net not being entirely destitute of atmosphere and clouds ;t
and what adds greatly to the probability of this is the ap-
pearance of brilliant white spots at its poles, — one of which

* I have noticed the phenomena described in the text on many occa-
sions, but never more distinct than on the occasion when the drawing
was made from whicli the figure in plate I. is engraved. — Author.

t It has been surmised to have a very extensive atmosphere, but on no
sufficient or even plausible grounds.

CHAP. Vin.3 APPEARANCES OF THE PLANETS. 265

appears in our figure, — which have been conjectured with
a great deal of probability to be snow ; as they disappear
when they have been long exposed to the sun, and are great-
est when just emerging from the long night of their polar
winter. By watching the spots daring a whole night,
and on successive nights, it is found that Mars has a ro-
tation on an axis inclined about 30° 18' to the ecliptic,
and in a period of 24*^ SO"" 2V in the same direction as the
earth's, or from west to east. The greatest and least appa-
rent diameters of Mars are 4" and 18", and its real dia-
meter about 4100 miles.

(438.) We come now to a much more magnificent pla-
net, Jupiter, the largest of them all, being in diameter no
less than 87,000 miles, and in bulk exceeding that of the
Earth nearly 1300 times. It is, moreover, dignified by
the attendance of four moons, satellites, or secondary
■planets, as they are called, which constantly accompany
and revolve about it, as the moon does round the earth,
and in the same direction, forming with their principal,
or pri7nary, a beautiful miniature system, entirely analo-
gous to that greater one of which their central body is
itself a member, obeying the same laws, and exemplifying,
in the most striking and instructive manner, the preva-
lence of the gravitating power as the ruling principle of
their motions : of these, however, we shall speak more
at large in the next chapter.

(439.) The disc of Jupiter is always observed to be
crossed in one certain direction by dark bands or belts,
presenting the appearance in plate I. Jig. 2, which repre-
sents this planet as seen on the 23d of September, 1832,
in the 20-feet reflector at Slough. These belts are, how-
ever, by no means alike at all times ; they vary in breadth
and in situation on the disc (though never in their general
direction). They have even been seen broken up, and
distributed over the whole face of the planet : but this
phenomenon is extremely rare. Branches running out
from them, and subdivisions, as represented in the figure,
as well as evident dark spots, like strings of clouds, are
by no means uncommon ; and from these, attentively
watched, it is concluded that this planet revolves in the
surprisingly short period of 9'' 55"" 50' (sid. time), on an
axis perpendicular to the direction of the belts. Now, it

Z

266 A TREATISE ON ASTRONOMV. [cHAP. VIII.

is very remarkable, and forms a most satisfactory com-
ment on the reasoning by which the spheroidal figure of
the earth has been deduced from its diurnal rotation, that
the outline of Jupiter's disc is evidently not circular, but
elliptic, being considerably flattened in the dii'ection of its
axis of rotation. This appearance is no optical illusion,
but is authenticated by micrometrical measures, which
assign 107 to 100 for the proportion of the equatorial and
polar diameters. And to confirm, in the strongest man-
ner, the truth of those principles on which our former
conclusions have been founded, and fully to authorize
their extension to this remote system, it appears, on calcu-
lation, that this is really the degree of oblateness which
corresponds, on those principles, to the dimensions of Ju-
piter, and to the time of his rotation.

(440.) The parallelism of the belts to the equator of
Jupiter, their occasional variations, and the appearances
of spots seen upon them, render it extremely probable
that they subsist in the atmosphere of the planet, forming
tracts of comparatively clear sky, determined by currents
analogous to our trade-winds, but of a much more steady
and decided character, as might indeed be expected from
the immense velocity of its rotation. That it is the
comparatively darker body of the planet which appears
in the bells is evident from this, — that they do not come
up in all their strength to the edge of the disc, but fade
away gradually before they reach it. (See plate I.
jig. 2.) The apparent diaixieter of Jupiter varies from
30" to 46".

(441.) A still more wonderful, and, as it may be
termed, elaborately artificial mechanism, is displayed in
Saturn, the next in order of remoteness to Jupiter, to which
it is not much inferior in magnitude, being about 79,000
miles in diameter, nearly 1000 times exceeding the earth
in bulk, and subtending an apparent angular diameter at
the earth, of about 16". This stupendous globe, be-
sides being attended by no less than seven satellites or
moons, is surrounded with two broad, flat, extremely
thin rings, concentric with the planet and with each
other; both lying in one plane, and separated by a very
narrow interval from each other throughout their whole
circumference, as they are from the planet by a much

CHAP. vin.J OF Saturn's rings. 267

wider. The dimensions of this extraordinary appendage

are as follows :* —

Miles.

Exterior diameter of exterior ring = 176418.

Interior ditto = 155272.

Exterior diameter of interior ring = 151690.

Interior ditto = 117339.

Equatorial diameter of the body = 79160.

Interval between the planet and interior ring = 19090.

Interval of the rings = 1791.

Thickness of the rings not exceeding = 100.

The figure {fig- 3, plate I.) represents Saturn surrounded
by its rings, and having its body striped with dark belts,
somewhat similar, but broader and less strongly marked
than those of Jupiter, and owing, doubtless, to a similar
cause. That tlie ring is a solid opake substance is shown
by its throwing its shadow on the body of the planet,
on the side nearest the sun, and on the other side re-
ceiving that of the body, as shown in the figure. From
the parallelism of the belts with the plane of the ring,
it may be conjectured that the axis of rotation of the
planet is perpendicular to that plane ; and this conjec-
ture is confirmed by the occasional appearance of ex-
tensive dusky spots on its surface, which wlien watched,
like the spots on Mars or Jupiter, indicate a rotation in
lO*" 29™ 17^ about an axis so situated.

(442.) The axis of rotation, like that of the earth,
preserves its parallelism to itself during the motion of
the planet in its orbit ; and the same is also the case
with the ring, whose plane is constantly inclined at the
same, or very nearly the same, angle to that of the orbit,
and, therefore to the ecliptic, viz. 28° 40' ; and intersects
the latter plane in a line, which makes an angle with the
line of equinoxes of 170°. So that the nodes of the
ring lie in 170° and 350° of longitude. Whenever, then,
the planet happens to be situated in one or other of these
longitudes, as at AB, the plane of the ring passes through
the sun, which then illuminates only the edge of it ;
and as, at the same moment, owing to the smallness of
the earth's orbit, E, compared with that of Saturn, the

* These dimensions are calculated from Prof. Struve's micrometric
mea-sures, Mem. Art. Soc. iii. 301, with the exception of the thickness of
the ring, which is concluded from my own observations, during its gra-
dual extinction now in progress. The interval of the rings here stated
is possibly somewhat too small.

268 A TREATISE ON ASTRONOMY. [cHAP. VIII.

oartli is necessarily not. far out of that plane, and must,
at all events, puss through it a little before or after that
moment, it only then appears to us a very fine straight
line, drawn across the disc, and projecting out on each
side — indeed, so very thin is the ring, as to be quite in-
visible, in this situation, to any but telescopes of extra-
ordinary power. This remarkable phenomenon takes
place at intervals of 15 years, but the disappearance of
the ring is generally double, the earth passing twice
through its plane before it is carried past our orbit by
the slow motion of Saturn. This second disappearance
is now in progress.* As the planet, however, recedes
from these points of its orbit, the line of sight becomes
gradually more and more inclined to the plane of the
ring, which, according to the laws of perspective, ap-
pears to open out into an ellipse which attains its greatest
breadth when the planet is 90° from either node, as at
CD. Supposing the upper part of the figure to be north,
and the lower south of the ecliptic, the north side only
of the ring will be seen when the planet lies in the
semicircle ACB, and the southern only when in ADB.
At the time of the greatest opening, the longer diameter
is almost exactly double the shorter.

(443.) It will naturally be asked how so stupendous
an arch, if composed of solid and ponderous materials,
can be sustained without collapsing and falling in upon
the planet? The answer to this is to be found in a swift
rotation of the ring in its own plane, which observation
has detected, owing to some portions of the ring being
a little less bright than others, and assigned its period at
10*' 29™ 17% which, from what we know of its dimen-
sions, and of the force of gravity in the Saturnian sys-
tem* is very nearly the periodic time of a satellite re-
volving at the same distance as the middle of its breadth.
It is the centrifugal force, then, arising from this rotation,
which sustains it; and, although no observation nice
enough to exhibit a difference of periods between the
outer and inner rings have hitherto been made, it is more
than probable that such a difierence does subsist as to

* The disappearance of tVio rings is complete, when observed with ri
reflector eighteen inches ia aperture, and twenty feet in focal length,
4prU 2i), mSi.— Author,

CHAP. viii.J OF Saturn's rings. 269

place each independently of the other in a similar state
of equilibrium.

(444.) Although the rings are, as we have said, very
nearly concentric with the body of Saturn, yet recent
micrometrical measurements of extreme delicacy have
demonstrated that the coincidence is not mathematically
exact, but that the centre of gravity of the rings oscillates
round that of the body describing a very minute orbit,
probal^ly under laws of much complexity. Trifling as
this remark may appear, it is of the \itmost importance
to the stability of the system of the rings. Supposing
them mathematically perfect in their circular form, and
exactly concentric Avith the planet, it is demonstrable
that they would form (in spite of their centrifugal force)
a system in a state of unstable equilibrium, which the
slightest external power would subvert — not by causing a
rupture in the substance of the rings- — but by precipita-
ting them, unbroken, on the surface of the planet. For
the attraction of such a ring or rings on a point or sphere
eccentrically situate within them, is not the same in all
directions, but tends to draw the point or sphere towards
the nearest part of the ring, or away from the centre.
Hence, supposing the body to become, from any cause,
ever so little eccentric to the ring, the tendency of their
mutual gravity is, not to correct but to increase this ec-
centricity, and to bring the nearest parts of them toge-
ther. (See chap. XI.) Now, external powers, capable
of producing such eccentricity, exist in the attractions
of the satellites, as will be shown in chap. XI. ; and in
order that the system may be stable, and possess within
itself a power of resisting the first inroads of such a ten-
dency, Avhile yet nascent and feeble, and opposing them
by an opposite or maintaining power, it has been shown
that it is sufficient to admit the rings to be loaded in some
part of their circumference, either by some minute in-
equality of thickness, or by some portions being denser
than others. Such a load would give to the whole ring
to which it Avas attached somewhat of the character of a
heavy and sluggish satellite, maintaining itself in an
orbit with a certain energy sufficient to overcome minute
causes of disturbance, and establish an average bearing
on its centre. But even without supposing the existence

z3

270 A TREATISE ON ASTRONOMY. [cHAP. VIII.

of any such load, — of which, after all, we have no
proof, — and granting, therefore, in its full extent, the
general instability of the equilibrium, we think we per-
ceive, in the periodicity of all the causes of disturbance,
a sufHcient guarantee of its preservation. However
homely be the illustration, we can conceive nothing more
apt in every way to give a general conception of this
maintenance of equilibrium under a constant tendency
to subversion, than the mode in which a practised hand
will sustain a long pole in a perpendicular position rest-
ing on the finger by a continual and almost imperceptible
variation of the point of support. Be that, however, as
it may, the observed oscillation of the centres of the rings
about that of the planet is in itself the evidence of a
perpetual contest between conservative and destructive
powers — both extremely feeble, but so antagonizing one
another as to prevent the latter from ever acquiring an
uncontrollable ascendancy, and rushing to a catastrophe.
(445.) This is also the place to observe, that, as the
smallest difference of velocity between the body and rings
must infallibly precipitate the latter on the former, never
more to separate (for they would, once in contact, have
attained a position of stable equiUbrium, and be held to-
gether ever after by an immense force) : it follows, either
that their motions in their common orbit round the sun
must have been adjusted to each other by an external
power, with the minutest precision, or that the rings must
have been formed about the planet Avhile subject to their
common orbitual motion, and under the full and free in-
fluence of all the acting forces.

(446.) The rings of Saturn must present a magnificent
spectacle from those regions of the planet which lie above
their enlightened sides, as vast arches spanning the sky
from horizon to horizon, and holding an invariable situa-
tion among the stars. On the other hand, in the regions
beneath the dark side, a solar eclipse of fifteen years in
duration, under their shadow, must aflbrd (to our ideas)
an inhospitable asylum to animated beings, ill compen-
sated by the faint light of the satellites. But we shall do
wrong to judge of the fitness or unfitness of their con-
dition from what we see around us, when, perhaps, the
very combinations which convey to our minds only im»

CHAP, vni.] GENERAL VIEAV OF THE SOLAR SYSTEM. 271

ages of horror, may bo in reality tlioatre.s of llie most
striking and glorious displays of beneficent contrivance.

(447.) Df Uranus we see nothing but a small, round,
uniformly illuminated disc, without rings, belts, or dis-
cernible spots. Its apparent diameter is about 4", from
which it never varies much, owing to the smallness of
our orbit in comparison of its own. Its real diameter is
about 35,000 miles, and its bulk 80 times that of the
earth. It is attended by satellites — two at least, probably
five or six — whose orbits (as Avill be seen in the next
chapter) offer remarkable peculiarities.

(448.) If the immense distance of Uranus precludes
all hope of coming at much knowledge of its physical
state, the minuteness' of the four ultra-zodiacal planets
is no less a bar to any inquiry into theirs. One of them,
Pallas, is said to have somewhat of a nebulous or hazy
appearance, indicative of an extensive and vaporous at-
mosphere, little repressed and condensed by the inade-
quate gravity of so small a mass. No doubt the most
remarkable of their peculiarities must lie in this condi-
tion of their state. A man placed on one of them would
spring with ease GO feet high, and sustain no greater
shock in his descent that he does on the earth from leap-
ing a yard. On such planets giants might exist ; and
those enormous animals, which on earth require the buoy-
ant power of water to counteract their weight, might
there be denizens of the land. But of such speculation
there is no end.

(449.) We shall close this chapter with an illustration
calculated to convey to the minds of our readers a gene-
ral impression of the relative magnitudes and distances
of the parts of our system. Choose any well levelled
field or bowling green. On it place a globe, two feet m
diameter ; this Avill represent the Sun ; Mercury will be
represented by a gi-ain of mustard seed, on the circum-
ference of a circle 164 feet in diameter for its orbit;
Venus a pea, on a circle 284 feet in diameter; the Earth
also a pea, on a circle of 430 feet ; Mars a rather largo
pin's head, on a circle of 654 feet ; Juno, Ceres, Vesta,
and Pallas, grains of sand, in orbits of from 1000 to 1200
feet ; Jupiter a moderate-sized orange, in a circle nearly
half a mile across ; Saturn a small orange, on a circle of

272 A TREATISE ON ASTRONOMY. [cHAP. IX.

four-fifths of a mile ; and Uranus a full-sized cherry, or
small plum, upon the circumference of a circle more than
a mile and a half in diameter. As to getting correct no-
tions on this subject by drawing circles on paper, or,
still worse, from those very childish toys called orreries,
it is out of the question. To imitate the motions of the
planets, in the above-mentioned orbits. Mercury must
describe its own diameter in 41 seconds ; Venus, in 4"
14'; the earth, in 7 minutes ; Mars, in 4'" 48'; Jupiter,
in 2" 56" ; Saturn, in 3" 13'" ; and Uranus, in 2'' IG".

CHAPTER IX.

OF THE SATELLITES.

Of the Moon, as a Satellite of the Earth — General Proximity of Satellites

to their Primaries, and consequent Subordination of their Motions

Masses of the Primaries concluded from the Periods of their Satellites
— Maintenance of Kepler's Laws in Ihe secondary Systems — Of Jupi-
ter's Satellites — Their Eclipses, &c. — Velocity of Light discovered by
their Means — Satellites of Saturn — Of Uranus.

(450.) In the annual circuit of the eartli about the sun,
it is constantly attended by its satellite the moon, which
revolves round it, or rather both round their common
centre of gravity; Avhile this centre, strictly speaking,
and not either of the two bodies thus connected, moves
in an elliptic orbit, undisturbed by their mutual action,
just as the centre of gravity of a large and small stone
tied together and flung into the air describes a parabola
as if it were a real material substance under the earth's
attraction, while the stones circulate round it or round
each other, as we choose to conceive the matter.

(451.) If we trace, therefore, the real curve actually
described by either the moon's or the earth's centres, in
virtue of this compound motion, it will appear to be, not
an exact ellipse, but an undulated curve, like that repre-
sented in the figure to article 272, only that the number
of undulations in a whole revolution is but 13, and their
actual deviation from the general ellipse, which serves
them as a central line, is comparatively very much smaller;
so much so, indeed, that every part of the curve described

CHAP, IX.] OF THE SATELLITES. 273

by cither the earth or moon is concave towards the sun.
The excursions of the earth on either side of the ellipse,
indeed, are so very small as to be hardly appreciable. In
fact, the centre of gravity of the earth and moon lies al-
ways within the surface of the earth, so that the monthly
orbit described by the earth's centre about the common
centre of gravity is comprehended within a space less
than the size of the earth itself. The effect is, neverthe-
less, sensible, in producing an apparent monthly dis-
placement of the sun in longitude, of a parallactic kind,
which is called the menstrual equation ; Avhose greatest
amount is, however, less than the sun's horizontal paral-
lax, or than 8*6".

(452.) The moon, as we have seen, is about 60 radii
of the earth distant from the centre of the latter. Its
proximity, therefore, to its centre of attraction, thus esti-
mated, is much greater than that of the planets to the
sun ; of which, Mercury, the nearest, is 84, and Uranus
2026 solar radii from its centre. It is owing to this prox-
imity that the moon remains attached to tlie earth as a
satellite. Were it much farther, the feebleness of its
gravity towards the earth would be inadequate to produce
that alternate acceleration and retardation in its motion
about the sun, which divests it of the character of an in-
dependent planet, and keeps its movements subordinate
to those of the earth. The one would outrun, or be left
behind the other, in their revolutions round the sun (by
reason of Kepler's third law), according to the relative
dimensions of their heliocentric orbits, after which the
whole influence of the earth would be confined to pro-
ducing some considerable periodical disturbance in the
moon's motion, as it passed or was passed by it in each
synodical revolution.

(453.) At the distance at which the moon really is
from us, its gravity towards the earth is actually less than
towards the sun. That this is the case, appears suffi-
ciently from what we have already stated, that the moon's
real path, even when between the earth and sun, is con-
cave towards the latter. But it will appear still more
clearly if, from the known periodic times* in which the

* R and r radii of two orbits (supposed circular), P and p the periodic

R r

times ; then the arcs in question (A and a) are to each other as — to - ;

274 A TREATISE ON ASTRONOMY. [cHAP. IX.

earth completes its annual and the moon its monthly orbit,
and from the dimensions of those orbits, we calculate the
amount of deflection, in either, from their tangents, in
equal very minute portions of time, as one second.
These are the versed sines of the arcs described in that
time in the two orbits, and these are the measures of the
acting forces which produce these deflections. If we
execute the numerical calculation in the case before us,
we shall find 2-209 : 1 for the proportion in which the
intensity of the force which retains the earth in its orbit
round the sun actually exceeds that by which the moon
is retained in its orbit about the earth.

(454.) Now the sun is 400 times more remote from
the earth than the moon is. And, as gravity increases as
the squai'es of the distances decrease, it must follow that,
at equal distances, the intensity of solar would exceed
that of terrestrial gravity in the above proportion, aug-
mented in the further ratio of the square of 400 to 1 ;
that is, in the proportion of 354936 to 1 ; and therefore,
if we grant that the intensity of the gravitating energy is
commensurate with the mass or inertia of the attracting
body, we are compelled to admit the mass of the earth
to be no more than 3-4V3-6- of that of the sun.

(455.) The argument is, in fact, nothing more than a
recapitulation of what has been adduced in chap. VII.
(art. 380.) But it is here re-introduced, in order to show
how the mass of a planet which is attended by one or
more satellites can be as it were weidied against the sun,
provided we have learned from observation the dimen-
sions of the orbits described by the planet about the sun,
and by the satellites about the planet, and also the periods
in which these orbits are respectively described. It is
by this method that the masses of Jupiter, Saturn, and
Uranus have been ascertained. (See Synoptic Table.)

(456.) Jupiter, as already stated, is attended by four
satellites, Saturn by seven ; and Uranus certainly by two,
and perhaps by six. These, with their respective pri-
maries (as the central planets are called), form in each

and since the versed sines are as the squares of the arcs directly and the

R r
radii iuverselv, these arc to each other as -— to — „ ; and in this ratio are

the forces actinic on the revolving bodies in either case.

CHAP. IX.] OF THE SATELLITES. 275

case miniature systems, entirely analogous, in the ge-
neral laws by which their motions are governed, to the
great system in which the sun acts the part of the pri-
mary, and the planets of its satellites. In each of these
systems the laws of Kepler are obeyed, in the sense,
that is to say, in Avhich they are obeyed in the planetary
system— approximately, and without prejudice to the
effects of mutual perturbation, of extraneous interference,
if any, and of that small but not imperceptible correction
which arises from the elliptic form of the central body.
Their orbits are circles or ellipses of very moderate ec-
centricity, the primary occupying one focus. About this
they describe areas very nearly proportional to the times ;
and the squares of ths periodical times of all the satellites
belonging to each planet are in proportion to each other
as the cubes of their distances. The tables at the end
of the volume exhibit a synoptic view of the distances
and periods in these several systems, so far as they are
at present known ; and to all of them it will be observed
that the same remark respecting their proximity to their
primaries holds good, as in the case of the moon, with a
similar reason for such close connexion.

(457.) Of these systems, however, the only one
which has been studied with great attention is that of
Jupiter; partly on account of the conspicuous brilliancy
of its four attendants, which are large enough to offer
visible and measurable discs in telescopes of great pow-
er; but more for the sake of their eclipses, which, as
they happen very frequently, and are easily observed,
aflbrd signals of considerable use for the determination
of terrestrial longitudes (art. 218). This method, in-
deed, until thrown into the back ground by the greater
facility and exactness now attainable by lunar observa-
tions (art. 219), was the best, or rather the only one
which could be relied on for great distances and long in-
tervals.

(458.) The satellites of Jupiter revolve from west to
east (following the analogy of the planets and moon), in
planes very nearly, although not exactly, coincident with
that of the equator of the planet, or parallel to its belts.
This latter plane is inclined 3° 5' 30" to the orbit of the
planet, and is therefore but little different from the plane

276 A TREATISE ON ASTRONOMY. [cHAP. IX.

of the ecliptic. Accordingly, we sec their orbits pro-
jected very nearly into straight lines, in which they ap-
pear to oscillate to and fro, sometimes passing before
Jupiter, and casting shadows on his disc (which are
very visible in good telescopes, like small round ink
spots), and sometimes disappearing behind the body, or
being eclipsed in its shadow at a distance from it. It is
by these eclipses that we are furnished with accurate
data for the construction of tables of the satellites' mo-
tions, as well as with signals for determining difTerences
of longitude.

(459.) The eclipses of the satellites, in their general
conception, are perfectly analogous to those of the moon,
but in their detail they differ in several particulars.
Owing to the much greater distance of Jupiter from the
sun, and its greater magnitude, the cone of its shadow or
umbra (art. 355) is greatly more elongated, and of far
greater dimension, than that of the earth. The satel-
lites are, moreover, much less in proportion to their
primary, their orbits less inclined to its ecliptic, and of
(comparatively) smaller dimensions, than is the case with
the moon. Owing to these causes, the three interior
satellites of Jupiter pass through the shadow, and are
totally eclipsed, every revolution ; and the fourth, though,
from the greater inclination of its orbit, it sometimes
escapes eclipse, and may occasionally graze as it were
the border of the shadow, and suffer partial eclipse, yet
this is comparatively rare, and, ordinarily speaking, its
eclipses happen, like those of the rest, each revolution.

(460.) These eclipses, moreover, are not seen, as is
the case with those of the moon, from the centre of their
motion, but from a remote station, and one Avhose situa-
tion with respect to the line of shadow is variable.
This, of course, makes no difference in the times of the
eclipses, but a very great one in their visibility, and in
their apparent situations with respect to the planet at the
moment of their entering and quitting the shadow.

(461.) Suppose S to be the sun, E the earth in its
orbit EFGK, J Jupiter, and at the orbit of one of its
satellites. The cone of the shadow, then, will have its
vertex at X, a point far beyond the orbits of all the sa-
tellites ; and the penumbra, owing to the great distance

CHAP. IX.] ECLIPSES OF JUPITEr's SATELLITES. 277

of the sun, and the consequent smaUness of the angle its
disc subtends at Jupiter, will hardly extend, within the

limits of the satellites' orbits, to any perceptible distance
beyond the shadow, — for which reason it is not repre-
sented in the lijrure. A satellite revolvinff from west to
east (in the direction of the arrows) will be eclipsed
when it enters the shadow at o, but not suddenly, be-
cause, like the moon, it has a considerable diameter seen
from tlie planet ; so that the time elapsing from the first
perceptible loss of light to its total extinction will be that
which it occupies in describing about Jupiter an angle
equal to its apparent diameter as seen from the centre
of the planet, or ratlier somew^hat more, by reason of the
penumbra ; and tlie same remark applies to its emer-
gence at b. Now, owing to the difference of telescopes
and of eyes, it is not possible to assign the precise mo-
ment of incipient obscuration, or of total extinction at a,
nor that of the first glimpse of light falling on the satel-
lite at b, or the complete recovery of its light. The ob-
servation of an eclipse, then, in which only the immer-
sion, or only the emersion, is seen, is incomplete, and
inadequate to afford any precise information, theoretical
or practical. But, if both the immersion and emersion
can be observed ivith the same telescope, and by the
same person, the interval of the times will give the du-
ration, and their mean the exact middle of the eclipse,
when the satellite is in the line SJX, i. e. the true mo-
ment of its opposition to the sun. Such observations,
and such only, are of use for determining the periods and
other particulars of the motions of the satellites, and for
affording data of any material use for the calculation of
terrestrial longitudes. The intervals of the eclipses, it

2 A

278 A TREATISE ON ASTRONOMY. []CHAP. IX

will be observed, give the synodic periods of the satel-
lites' revolutions ; from which their sidereal periods must
be concluded by the method in art. 353 (note).

(462.) It is evident, from a mere inspection of our
figure, that the eclipses take place to the west of the
planet, when the earth is situated to the west of the line
SJ, i. e. before the opposition of Jupiter ; and to the
east, when in the other half of its orbit, or after the op-
position. When the earth approaches the opposition, tlie
visual line becomes more and more nearly coincident
with the direction of the shadow, and the apparent
place where the eclipses happen will be continually
nearer and nearer to the body of the planet. When the
earth comes to F, a point determined by drawing- bY to
touch the body of the planet, the emersions will cease
to be visible, and will thenceforth, to an equal distance
on the other side of the opposition, happen behind the
disc of the planet. When the earth arrives at G (or H)
the immersion (or emersion) will happen at the very
edge of the visible disc, and when between G and H (a
very small space) the satellites will jmss unedipsed be-
hind the limb of the planet.

(463.) When the satellite comes to m, its shadow will
be thrown on Jupiter, and will appear to move across it
as a black spot till the satellite comes to n. But the satel-
lite itself Avill not appear to enter on the disc till it comes
up to the line drawn from E to the eastern edge of the
disc, and will not leave it till it attains a similar line
drawn to the western edge. It appears then that the
shadow Avill precede the satellite in its progress over the
disc before the opposition, and vice versa. In these
transits of the satellites, which, with very powerful
telescopes, may be observed with great precision, it fre-
quently happens that the satellite itself is discernible on
the disc as a bright spot if projected on a dark belt ; but
occasionally also as a dark spot of smaller dimensions
than the shadow. This curious fact (observed by Schroe-
ter and Harding) has led to a conclusion that certain
of the satellites have occasionally on their own bodies,
or in their atmospheres, obscure spots of great extent.
We say of great extent; for the satellites of Jupiter,
small as they appear to us, are really bodies of con-

CHAP. IX.] OF Jupiter's satellites.

279

siderable size, as the following comparative table will
show.*

Mean apparent
diameter.

Diameter in
miles.

Mass.t

Jupiter
1st satellite

3ti"-3-27
1-105
0-911
1-488
1-273

87000
2508
2008
3377
2890

lOOOOOOO
0 0000173
00000232
00000885
0-0000427

■-).(

4th

(464.) An extremely singular relation subsists be-
tween the mean angular velocities or 'mean motions (as
they are termed) of the three first satellites of Jupiter.
If tlie mean angular velocity of the first satellite be added
to twice that of the third, the sum will equal three times
that of the second. From this relation it follows, that if
from the mean longitude of the first added to twice that
of the third, be subducted three times that of the second,
tlie remainder will always be the same, or constant, and
observation informs us that this constant is 180°, or two
right angles ; so that, the situations of any two of them
being given, that of the third may be found. It has been
attempted to account for this remarkable fact, on the
theory of gravity by their mutual action. One curious
consequence is, that these three satellites cannot be all
eclipsed at once ; for, in consequence of the last-men-
tioned relation, when the second and tliird lie in the
same direction from the centre, the first must lie on the
opposite; and therefore, when the first is eclipsed, the
other two must lie between the sun and planet, throwing
its shadow on the disc, and vice versa. One instance only
(so far as we are aware) is on record when Jupiter has
been seen vAthout satellites ; viz. by Molyneux, Nov.
2 (old style), 1681.^

(465.) The discovery of Jupiter's satellites by Galileo,
one of the first-fruits of the invention of the telescope,
forms one of the most memorable epochs in the history
of astronomy. The first astronomical solution of the
great problem of " the longitude'''' — the most important
for the interests of mankind which has ever been brought

o

under the dominion of strict scientific principles, dates

* Struve, Mem. Ast. Soc. iii. 301. t Laplace, Mec. Col. liv. viii. % 27.
X Molyneux, Optics, p. 271.

280 A TREATISE ON ASTRONOMY. [cHAP. IX.

immediately from their discovery. The final and con-
clusive establishment of the Copernican system of as-
tronomy may also be considered as referable to the dis-
covery and study of this exquisite miniature system, in
which the laws of the planetary motions, as ascertained
by Kepler, and especially that which connects their
periods and distances, were speedily traced, and found
to be satisfactorily maintained. And (as if to accumulate
liistorical interest on this point) it is to the observation of
their eclipses that we owe the grand discovery of the
aberration of light, and the consequent determination of
the enormous velocity of that wonderful element. This
we must explain now at large.

(466.) The earth's orbit being concentric with that of
Jupiter and interior to it (see Jig. art. 460), their mutual
distance is continually varying, the variation extending
from the sum to the difference of the radii of the two
orbits, and the difference of the greater and least dis-
tances being equal to a diameter of the earth's orbit.
Now, it was observed by Roemer (a Danish astronomer,
in 1675), on comparing together observations of eclipses
of the satellites during many successive years, that the
eclipses at and about the opposition of Jupiter (or its
nearest point to the earth) took place too soon — sooner,
that is, than, by calculation from an average, he expected
them ; whereas those which happened when the earth
was in the part of its orbit most remote from Jupiter
Avere always too late. Connecting the observed error in
their computed times with the variation of distance, he
concluded, that, to make the calculation on an average
period correspond with fact, an allowance in respect of
time behooved to be made proportional to the excess or
defect of Jupitei-'s distance from the earth above or below
its average amount, and such that a difference of distance
of one diameter of the earth's orbit should correspond to
16'" 26'*6 of time allowed. Speculating on the probable
physical cause, he was naturally led to think of the
gradual instead of an instantaneous propagation of light.
This explained every particular of the observed phe-
nomenon, but the velocity required (192000 miles per
second) was so great as to startle many, and, at all events,
to require confirmation. This has been afforded since,

CHAP. IX.] SUCCESSIVE TRANSMISSION OF LIGHT. 281

and of the most unequivocal kind, by Bradley's discovery
of the aberration of light (art. 275). The velocity of light
deduced from this last phenomenon differs by less than one
eightieth of its amount from that calculated from the
eclipses, and even this dilference will no doubt be de-
stroyed by nicer and more rigorously reduced observations.
(467.) The orbits of Jupiter's satellites are but little
eccentric ; those of the two interior, indeed, have no per-
ceptible eccentricity ; their mutual action produces in
them perturbations analogous to those of the planets
about the sun, and which have l)een diligently investi-
gated by Laplace and others. By assiduous observation
it has been ascertained that they are subject to marked
fluctuations in respect of brightness, and that these fluc-
tuations happen periodically, according to their position
with respect to the sun. From this it has been con-
cluded, apparently with reason, that they turn on their
axes, like our moon, in periods equal to their respective
sidereal revolutions about their primary.

(468.) The satellites of Saturn have been much less
studied than those of Jupiter. The most distant is by
far the largest, and is probably not much inferior to Mars
in size. Its orbit is also materially inclined to the plane
of the ring, witli which those of all the rest nearly coin-
cide. It is the only one of the number whose theory
has been at all inquired into, further than suflices to
verify Kepler's law of the periodic times, which holds
good, mutatis mutandis, and under the requisite reser-
vations, in this as in the system of Jupiter. It exhibits,
like those of Jupiter, periodic defalcations of light,
which prove its revolution on its axis in the time of a
sidereal revolution about Saturn. The next in order (pro-
ceeding inwards) is tolerably conspicuous ; the three next
very minute, and requiring pretty powerful telescopes to
see them ; while the two interior satellites, which just
skirt the edge of the ring, and move exactly in its plane,
have never been discerned but with the most powerful
telescopes which human art has yet constructed, and
then only under peculiar circumstances. At the time of
the disappearance of the ring (to ordinary telescopes)
they have been seen* tlyeading like beads the almost
* By my father, in 1789, with a reflecting telescope four feet in aperture.

2 a2

282 A TREATISE ON ASTRONOMY. [cHAP. IX.

infinitoly thin fibro of light to which it is then reduced, and
for a short time advancing olf it at either end, speedily to
return, and hastening to their habitual concealment.
Owing to the obliquity of the ring, and of the orbits of
the satellites to Saturn's ecliptic, there are no eclipses of
the satellites (the interior ones excepted) until near the
time when the ring is seen edgewise.

(469.) With the exception of the two interior satel-
lites of Saturn, the attendants of Uranus are the most dif-
ficult objects to obtain a sight of, of any in our system.
Two undoubtedly exist, and four more have been sus-
pected. These two, however, offer remarkable and, in-
deed, quite unexpected and unexampled peculiarities.
Contrary to the unbroken analogy of tlie whole planet-
ary system — whether of primaries or secondaries — the
planes of their orbits arc nearly perpendicular to the
ecliptic, being inclined no less than 78° 58' to that plane,
and in these orbits their motions are retrograde ; that is
to say, their positions, when projected on the ecliptic,
instead of advancing from west to east round the centre
of their primary, as is the case with every other planet
and satellite, move in the opposite direction. Their
orbits are nearly or quite circular, and they do not appear
to have any sensible, or, at least, any rapid motion of
nodes, or to have undergone any material change of incli-
nation, in the course, at least, of half a revolution of their
primary round the sun.*

* These anomalous peculiarities, which seem to occur at the extreme
limits of our system, as if to prepare us for further departure from all its
analogies, in other systems which may yet be disclosed to us, have hith-
erto rested on the sole testimony of their discoverer, who alone had ever
obtained a view of them. I am happy to be able, from my own observa-
tions li-om 1S28 to the present time, to confirm, in the amplest manner, my
father's results. — Author.

CHAP. X. I NUMBER OF COMETS. 283

CHAPTER X.

OF COMETS.

Great Number of recorded Comets— The number of unrecorded proba-
bly much greater— Description of a Comet— Comets without Tails-
Increase and Decay of their Tails— Their Motions— Subject to the
general Laws of planetary Motion— Elements of their Orbits— Periodic
Return of certain Comets— Halley's—Encke's—Biela's— Dimensions of
Comets— Tlicir Resistance by the Ether, gradual Decay, and iMssible
Dispersion in Space.

(470.) The extraordinary aspect of comets, their rapid
and seemingly irregular motions, the unexpected manner
in which they often burst upon us, and the imposing
magnitudes which they occasionally assume, have in all
ages rendered them objects of astonishment, not unmixed
with superstitious dread to the uninstructed, and an enig-
ma to those most conversant with the wonders of crea-
tion and the operations of natural causes. Even now,
that we have ceased to regard their movements as irregu-
lar, or as governed by other laws than those which retain
the planets in their orbits, their intimate nature, and the
offices they perform in the economy of our system, are
as much unknown as ever. No rational or even plausible
account has yet been rendered of those immensely volu-
minous appendages wliich they bear about with them,
and wliich are known by the name of their tails, (though
improperly, since they often precede them in their mo-
tions), any more than of several other singularities which
they present.

(471.) The number of comets which have been astro-
nomically observed, or of which notices have been re-
corded in history, is very great, amounting to several
hundreds ;* and when we consider that in the earlier ages
of astronomy, and indeed in more recent times, before the
invention of the telescope, only large and conspicuous

* See catalogues in the Almagest of Riecioli ; Pingre's Cometographia;
Delambre's Astron. vol. iii. ; Astronomische Abhandlungen, No. 1.
(which contains the elements of all the orbits of comets which have been
computed to the time of its publication, 1823) ; also, a catalogue now in
progress, by the Rev. T. J. liussey. Lon. & Ed. Phil. Mag. vol. ij. No. 9.
et seq. In a list cited by Lalande from the 1st vol. of the Tables de Ber-
lin, 700 comets are enumerated.

284 A TREATISE ON ASTRONOMY. [cHAP, X.

ones were noticed ; and tliat, since due attention has been
paid to the subject, scarcely a year has passed without
the observation of one or two of these bodies, and that
sometimes two and even three have appeared at once ; it
will be easily supposed that their actual number must be
at least many thousands. Multitudes, indeed, must es-
cape all observation, by reason of their paths traversing
only that part of the heavens which is above the horizon
in the daytime. Comets so circumstanced can only be-
come visible by the rare coincidence of a total eclipse of
the sun, — a coincidence which happened, as related by
Seneca, 60 years before Christ, when a large comet was
actually observed very near the sun. Several, however,
stand on record as having been bright enough to be seen
in the daytime, even at noon and in bright sunshine.
Such were the comets of 1402 and 1532, and that which
appeared a little before the assassination of C^aisar, and
was {afterwards') supposed to have predicted his death.
(472.) That feelings of awe and astonishment should
be excited b}; the sudden and unexpected appearance of
a great comet, is no way surprising ; being, in fact, ac-
cording to the accounts we have of such events, one of
the most brilliant and imposing of all natural phenomena.
Comets consist for the most part of a large and splendid
but ill defined nebulous mass of light, called the head,
which is usually much brighter towards the centre, and
offers the appearance of a vivid nucleus, like a star or pla-
net. From the head, and in a direction opposite to that
in which the sun is situated from the comet, appear to
diverge two streams of light, which grow broader and
more diffused at a distance from the head, and which
sometimes close in and unite at a little distance behind
it, sometimes continue distinct for a great part of their
course ; producing an effect like that of the trains left by
some bright meteors, or like the diverging fire of a sky-
rocket (only without sparks or perceptible motion). This
is the tail. This magnificent appendage attains occasion-
ally an immense apparent length. Aristotle relates of the
tail of the comet of 371 a. c, that it occupied a third of
the hemisphere, or 60° ; that of A. d. 1618 is stated to
have been attended by a train no less than 104° in length.
The comet of 1680, the roost celebrated of modern times,

CHAP. X.] SMALL DENSITY OF COMETS. 285

and on many accounts the most remarkable of all, with a
head not exceeding in brightness a star of the second
magnitude, covered with its tail an extent of more than
70° of the heavens, or, as some accounts state, 90°. The
figure {fig. 2, plate II.) is a very correct representation
of the comet of 1819 — by no means one of the most con-
siderable, but the latest which has been conspicuous to
the naked eye.

(473.) The tail is, however, by no means an invariable
appendage of comets. Many of the brightest have been
observed to have short and feeble tails, and not a few have
been entirely without them. Those of 1585 and 1763
offered no vestige of a tail ; and Cassini describes the
comet of 1682 as being as round and as bright as Jupiter.
On the other hand, instances are not wanting of comets
furnished with many tails or streams of diverging light.
That of 1744 had no less than six, spread out like an im-
mense fan, extending to a distance of nearly 30° in length.
The tails of comets, too, are often curved, bending, in
general, towards the region which the comet has left, as
if moving somewhat more slowly, or as if resisted in their
course.

(474.) The smaller comets, such as are visible only in
telescopes, or with difficulty by the naked eye, and which
are by far the most numerous, offer very frequently no
appearance of a tail, and appear only as round or some-
what oval vaporous masses, more dense towards the cen-
tre, where, however, they appear to have no distinct nu-
cleus, or any thing which seems entitled to be considered
as a solid body. Stars of the smallest magnitude remain .
distinctly visible, though covered by what appears to be
the densest portion of their substance ; although the same
stars would be completely obliterated by a moderate fog,
extending only a few yards from the surface of the earth.
And since it is an observed fact, that even those larger
comets which have presented the appearance of a nu-
cleus have yet exhibited no phases, though we cannot
doubt that they shine by the reflected solar light, it fol-
lows that even these can only be regarded as great masses
of thin vapour, susceptible of being penetrated through
their whole substance by the sunbeams, and reflecting
them alike from their interior parts and from their sur-

286 A TREATISE ON ASTRONOMY. [cHAP. X.

faces. Nor will any one regard this explanation as
forced, or feel disposed to resort to a phosphorescent qua
lity in tlie comet itself, to account for the phenomena in
question, wlien we consider (what will be hereafter
shown) the enormous magnitude of the space thus illumi-
nated, and the exti'emely small mass which there is
ground to attribute to these bodies. It will then be evi-
dent that the most vmsubstantial clouds which float in the
highest regions of our atmosphere, and seem at sunset to
be drenched in light, and to glow throughout their whole
depth as if in actual ignition, without any shadow or dark
side, must he looked upon as dense and massive bodies
compared with the filmy and all but spiritual texture of
a comet. Accordingly, whenever powerful telescopes
have been turned on these bodies, they have not failed to
dispel the illusion which attributes solidity to that more
condensed part of the head, which appears to the naked
eye as a nucleus ; though it is true that in some, a very
minute stellar point has been seen, indicating the exist-
ence of a solid body.

(475.) It is in all probability to the feeble coercion of
the elastic power of their gaseous parts, by the gravitation
of so small a central mass, that we must attribute this ex-
traordinary developement of the atmospheres of comets.
If the earth, retaining its present size, were reduced, by
any internal change (as by hollowing out its central
parts) to one thousandth part of its actual mass, its
coercive power over tlie atmosphere would be dimi-
nished in the same proportion, and in consequence the
latter would expand to a thousand times its actual bulk ;
and indeed much more, owing to the still farther dimi-
nution of gravity, by the recess of the upper parts from
the centre.

(476.) That the luminous part of a comet is something
in the nature of a smoke, fog, or cloud, suspended in a
transparent atmosphere, is evident from a fact which has
been often noticed, viz. — that the portion of the tail
where it comes up to, and surrounds the head, is yet
separated from it by an interval less luminous, as if sus-
tained and kept off from contact by a transparent stratum,
as we often see one layer of clouds laid over another
with a considerable clear space between. These, and

CHAP. X.] MOTIONS OF COMETS. 287

most of the other facts observed in the history of comets,
appear to indicate that the structure of a comet, as seen
in section in the direction of its length, must be that of
a hollow envelope, of a parabolic form, enclosing near its
vertex the nucleus and head, something as represented
in the annexed figure. This would account for the ap-

(©>•

parent division of the tail into two principal lateral
branches, the envelope being oblique to the line of sight
at its borders, and therefore a greater depth of illumi-
nated matter being there exposed to the eye. In all proba-
bility, however, they admit great varieties of structure,
and among them may very possibly be bodies of widely
different physical constitution.

(477.) We come now to speak of the motions of co-
mets. These are apparently most irregular and capri-
cious. Sometimes they remain in sight for only a few-
days, at others for many mouths; some move with ex-
treme slowness, others with extraordinary velocity ;
while not unfrequeutly, the two extremes of apparent
speed are exhibited by the same comet in diiferent parts
of its course. The comet of 1472 described an arc of
the heavens of 120° in extent in a single day. Some
pursue a direct, some a retrograde, and others a tortuous
and very irregular course : nor do they confine them-
selves, like the planets, within any certain region of the
heavens, but traverse indifferently every part. Their
variations in apparent size, during the time they continue
visible, are no less remarkable than those of their velo-
city ; sometimes they make their first appearance as faint
and slow moving objects, with little or no tail ; but by
degrees accelerate, enlarge, and throw out from them this
appendage, which increases in length and brightness till
(as always happens in such cases) they approach the
sun, and are lost in his beams. After a time they again
emerge, on the other side, receding from the sir^ ^vUh a

288 A TREATISE ON ASTRONOMY. [cHAP. X.

velocity at first rapid, but gradually decaying. It is after
thus passing the sun, and not till then, that they shine
forth in all their splendour, and that their tails acquire
tlieir greatest length and developement ; thus indicating
plainly the action of the sun's rays as the exciting cause
of that extraordinary emanation. As they continue to
recede from the sun, their motion diminishes and the
tail dies away, or is absorbed into the head, which itself
grows continually feebler, and is at length altogether lost
sight of, in by far the greater number of cases never to
be seen more,

(478.) Without tlie clue furnished by the theory of
gravitation, the enigma of these seemingly irregular and
capricious movements might have remained for ever un-
resolved. But Newton, having demonstrated the pos-
sibility of any conic section whatever being described
about the sun, by a body revolving under the dominion
of that law, immediately perceived the applicability of
the general proposition to the case of cometary orbits,
and the great comet of 1680, one of tlie most remark-
able on record, both for the immense length of its tail
and for the excessive closeness of its approach to the
sun (within one sixth of the diameter of that luminary),
afforded him an excellent opportunity for the trial of his
theory. The success of the attempt was complete. He
ascertained that this comet described about the sun as its
focus an elliptic orbit of so great an eccentricity as to be
undistinguishable from a parabola (which is the extreme,
or limiting form of the ellipse when the axis becomes
infinite), and that in this orbit the areas described about
the sun were, as in the planetary ellipses, proportional
to the times. The representation of the apparent mo-
tions of this comet by sucli an orbit, throughout its whole
observed course, was found to be as complete as those
of the motions of the planets in their nearly circular
paths. From that time it became a received truth, that
the motions of comets are regidated by the same general
laws as those of the planets — tlie difference of the cases
consisting only in the extravagant elongation of their el-
lipses, and in the absence of any limit to the inclinations
of their planes to that of the ecliptic — or any general co-
incidence in the direction of the motions from west to

CHAP. X.] PERIODICAL COMETS. 289

east, rather than from east to west, like what is observed
among the planets.

(479.) It is a problem of pure geometry, from the
general laws of elliptic or parabolic motion, to find the
situation and dimensions of the ellipse or parabola which
shall represent the motion of any given comet. In ge-
neral, three complete observations of its right ascension
and declination, with the times at which they were
made, suffice for the solution of tliis problem (wliich is,
however, a very difficult one), and for the determination
of the elements of the orbit. These consist, mutatis
mutandis, of the same data as are required for the com-
putation of the motion of a jihuiet : and, once deter-
mined, it becomes very easy to compare them with the
whole observed course of the comet, by a process ex-
actly similar to that of art. 426, and thus at once to as-
certain their correctness, and to put to the severest trial
the truth of those general laws on which all such calcu-
lations are founded.

(480.) For the most part, it is found that the motions
of comets may be sufficiently well represented by para-
bolic orbits, — that is to say, ellipses whose axes are of
infinite length, or, at least, so very long that no appre-
ciable error in the calculation of their motions, during all
the time they continue visible, would be incurred by
supposing them actually infinite. The parabola is that
conic section which is tlie limit between the ellipse on
the one hand, whicli returns into itself, and the hyper-
bola on the other, which runs out to infinity. A comet,
therefore, which should describe an elliptic path, how-
ever long its axis, must have visited the sun before, and
must again return (unless disturbed) in some determinate
period, — but should its orbit be of the hyperbolic cha-
racter, when once it has passed its perihelion, it could
never more return within the sphere of our observation,
but must run off to visit other systems, or be lost in the
immensity of space. A very few comets have been as-
certained to move in hyperbolas, Init many more in
ellipses. These then, in so far as their orbits can remain
unaltered by the attractions of tlie planets, must be re-
garded as permanent members of our system,

(481.) The most remarkable of these is the comet of

2B

290 A TREATISE ON ASTRONOMY. [cHAP. X.

Halley, so called from the celebrated Edmund Halley,
who, on calculating its elements from its perihelion pas-
sage in 1682, when it appeared in great splendour, with
a tail 30° in length, was led to conclude its identity with
the great comets of 1531 and 1607, Avhose elements he
had also ascertained. The intervals of these successive
apparitions being 75 and 76 years, Halley was encou-
raged to predict its re-appearance about the year 1759.
So remarkable a prediction could not fail to attract the
attention of all astronomers, and, as the time approached,
it became extremely interesting to know whether the at-
tractions of the larger planets might not materially inter-
fere with its orbitual motion. The computation of their
influence from the Newtonian law of gravity, a most
difficult and intricate piece of calculation, was undertaken
and accomplished by Clairaut, who found that the action
of Satuni would retard its retui-n by 100 days, and that
of Jupiter by no less than 518, making in all 618 days,
by which the expected return would happen later than
on the supposition of its retaining an unaltered period —
and that, in short, the time of the expected perihelion
passage would take place within a month, one way or
other, of the middle of April, 1759. — It actually hap-
pened on the 12th of March in that year. Its next re-
turn to the perihelion has been calculated by Messrs.
Damoiseau and Pontecoulant, and fixed by the former
on the fourth, and by the latter on the seventh of Novem-
ber, 1835, about a month or six weeks before which time
it may be expected to become visible in our hemisphere ;
and, as it will approach pretty near the earth, will very
probably exhibit a brilliant appearance, though, to judge
from the successive degradations of its apparent size and
the length of its tail in its several returns since its first
appearances on record (in 1305, 1456, &;c.), we are not
now to expect any of those vast and awful phenomena
which threw our remote ancestors of the middle ages into
agonies of superstitious terror, and caused public prayers
to be put up in the churches against the comet and its
malignant agencies.

(482.) More recently, two comets have been especially
identified as having performed several revolutions about
the sun, and as having been not only observed and re-

CHAP. X.] RESISTANCE EXPERIENCED BY COMETS. 291

corded in preceding revolutions, without knowledge of
this remarkable peculiarity, but have had already seve-
ral times their returns predicted, and have scrupulously
kept to their appointments. The first of these is the
comet of Encke, so called from Professor Encke, of Ber-
lin, who first ascertained its periodical return. It re-
volves in an ellipse of great eccentricity, inclined at an
angle of about 13° 22' to the plane of the ecliptic, and
in the short period of 1207 days, or about 31 years.
This remarkable discovery was made on the occasion of
its iourth recorded appearance, in 1819. From the el-
lipse then calculated by Encke, its return in 1822 was
predicted by him, and observed at Paramatta, in New
South Wales, by M. Rumker, being invisible in Europe :
since which it has been re-predicted, and re-observed in
all the principal observatories, both in the northern and
southern hemispheres, in 1825, 1828, and 1832, Its
next return will be in 1835.

(483.) On comparing the intervals between the suc-
cessive perihelion passages of this comet, after allowing
in the most careful and exact manner for all the disturb-
ances due to the actions of the planets, a very singular
fact has come to light, viz. that the periods are continu-
ally diminishing, or, in other words, the mean distance
from the sun, or the major axis of the ellipse, dwindling
by slow but regular degrees. This is evidently the effect
which would be produced by a resistance experienced
by the comet from a very rare ethereal medium pervading
the regions in which it moves ; for such resistance, by
diminishing its actual velocity, would diminish also its
centrifugal force, and thus give the sun more power over
it to draw it nearer. Accordingly (no other mode of
accounting for the phenomenon in question appearing),
this is the solution proposed by Encke, and generally
received. It will, therefore, probably fall ultimately into
the sun, should it not first be dissipated altogether — a
thing no way improbable, when the lightness of its ma-
terials is considered, and which seems authorized by the
observed fact of its having been less and less conspicuous
at each reappearance.

(484.) The other comet of short period which has
lately been discovered is that of Biela, so called from

292 A TREATISE ON ASTRONOMY. [cHAP. X.

M. Biela, of Joscphstadt, who first arrived at this inte-
resting conekision. It is identical with comets which
appeared in 1789, 1795, Sic, and describes its mode-
rately eccentric ellipse about the sun in 6| years ; and
the last apparition having taken place according to the
prediction in 1832, the next will be in 1838. It is a
small insignificant comet, without a tail, or any appear-
ance of a solid nucleus whatever. Its orbit, by a re-
markable coincidence, very nearly intersects that of the
earth ; and had the latter, at the time of its passage in
1832, been a month in advance of its actual place, it
would have passed through the comet — a singular ren-
contre, perhaps not unattended with danger.*

(485.) Comets in passing among and near the planets
are materially drawn aside from their courses, and in
some cases have their orbits entirely clianged. This is
remarkably the case with Jupiter, which seems by some
strange fatality to be constantly in their way, and to
serve as a perpetual stumbling block to them. In the
case of the remarkable comet of 1770, which was found
by Lexell to revolve in a moderate ellipse in the period
of about 5 years, and whose i"eturn was predicted by him
accordingly, the prediction was disappointed by the comet
actually getting entangled among the satellites of Jupiter,
and being completely throAvn out of its orbit by the at-
traction of that planet, and forced into a much larger el-
lipse. By this extraordinaiy rencontre, the motions of
the satellites suffered not the least perceptible derunp^e-
tnent — a sufficient proof of the smallness of the comet's
mass.

(486.) It I'emains to say a few words on the actual di-

* Should calculation eslabtish the fact of a resistince experienced also
by this comet, the subject of periodical c-omets will assume an extraor-
dinary degree of interest. It cannot be doubted that many more will
be discovered, and by their resistance questions will come to be decided,
such as the following : — What is the law of density of the resisting medium
wliich surrounds the sun ? Is it at rest or in motion I If the latter, in
what direction does it move ? Circularly round the sun, or ti-aversing
space? If circularly, in what plane? It is obvious that a circular or
vorticose motion of the ether would acceierate some comets and retard
others according as their revolution was, relative to such motion, direct
or retrtigrade. Su])[)osing the neighbourhood of the sini to bo filled with
a material fluid, it is not conceivable that the circulation of the planets
in it for ages should not have impressed upon it some degree of rotation
in their own direction. And this may preserve them from the extreme
effects of accumulated resistance. — Autbor.

CHAP. X.] DIMENSIONS OF COMETS. 293

niensions of comets. The calculation of the diameters
of their heads, and the lengths and breadths of their
tails, offers not the slightest difficulty when once the
elements of their orlnts are known, for by these we know
their real distances from the earth at any time, and the
true direction of the tail, which we see only foreshort-
ened. Now calculations instituted on these principles
lead to the sur})rising fact, that the comets are by far the
most voluminous bodies in our system. The following
are the dimensions of some of those which have been
made the subjects of such inquiry.

(487.) The tail of the great comet of 1680, imme-
diately after its perihelion passage, was found by New-
ton to have been no less than 20000000 of leagues in
length, and to have occupied only two days in its emis-
sion from the comet's body ! a decisive proof this of its
being darted forth by some active force, the origin of
which, to judge from the direction of the tail, must be
sought in the sun itself. Its greatest length amounted
to 41000000 leagues, a length much exceeding the
whole interval between the sun and earth. The tail of
the comet of 1769 extended 16000000 leagues, and that
of the great comet of 1811, 36000000. The portion of
the head of this last comprised within the transparent
atmospheric envelope which separated it from the tail
was 180000 leagues in diameter. It is hardly conceiv-
able that matter once projected to such enormous dis-
tances should ever be collected again by the feeble at-
traction of such a body as a comet — a consideration
which accounts for the rapid progressive diminution of
the tails of such as have been frequently observed,

(488.) A singular circumstance lias been remarked
respecting the change of dimensions of the comet of
Eneke in its progress to and retreat from the sun : viz.
that the real diameter of the visible nebulosity under-
goes a rapid contraction as it approaches, and an equally
rapid dilatation as it recedes from the sun. M. Valz,
who, among others, had noticed this fact, has accounted
for it by supposing a real compression or condensation
of volume, owing to the pressure of an ethereal medium
growing more dense in the sun's neighbourhood. It is
very possible, however, that the change may consist in

2 B 2

294 A TKEATIS5E ON ASTRONOMY. [ciIAP. XI.

no real expansion or condensation of volume (further
than is clue to the convergence or divergence of the dif-
ferent parabolas described by each of its molecules to or
from a common vertex), but may rather indicate the al-
ternate conversion of evaporable materials in the upper
regions of a transparent atmospliere, into the states of
visible cloud and invisible gas, by the mere effects of
heat and cold. But it is time to quit a subject so myste-
rious, and open to such endless speculation.

CHAPTER XL

OF PERTURBATIONS.

Subject propounded — Superposition of small Mutions — IVoblem of thrre
Bodies — Eslimation of disturbing Forces — Motion of Nudes — Changes
of Inclination — Compensation operated in a whole Revolution of the
Node — Lagrange's Theorem of the Stability of the Inclinations —
Change of Obhquity of the Eclifitic — Precession of the Equinoxes —
INutation — Tlieorem respecting forced Vibrations — Of ihe Tides — Va-
riation of Elements of the I'ianet's Orbits — Periodic and secular —
Disturbing Forces considered as tangential and radial — Eflects of tan-
gential Force — 1st, in circular Orbits; 2diy, in elliptic — Compensations
effected — Case of near Commensurability of mean Motions — The great
Inecpiality of Jupiter and Saturn explained — The long Ineipiality of
Venus and the Earth — Lunar Variation — Etlijcts of the radial Force —
Mean Effect on the Period and Dimensions of the disturbed Orbit —
Variable Part of its Effect — Lunar Eveclion — Secular Acceleration
of the Moon's Motion — Invariabilily of the Axes and Periods — Theoiy
of the secular Variations of the Eccentricities and Perihelia — Motion
of the lunar Apsides^Lagrange's Theorem of the Stability of Ihe
Eccentricities — Nutation oi' llie liniar Orbit — Perturbations of Jupi-
ter's Satellites.

(489.) In the progress of this work, we have more
than once called the reader's attention to the existence
of inequalities in the lunar and planetary motions not
included in the expression of Kepler's laws, but in some
sort supplementary to them, and of an order so far sub-
ordinate to those leading features of the celestial move-
ments, as to require, for their detection, nicer observa-
tions, and longer continued comparison between facts
and theories, than suffice for the establishment and veri-
fication of the elliptic theory. These inequalities are
known, in physical astronomy, by the name of pertur-

CHAP. XI. J OF PERTURBATIONS. 295

hations. Tliey arise, in the case of the primary planets,
from the mutual gravitations of these planets towards
each other, wliich derange their elliptic motions round
the sun ; and in that of the secondaries, partly from
tlie mutual gravitation of the secondaries of the same
system similarly deranging their elliptic motions round
their common primary, and partly from the unequal
attraction of the sun on them and on their primary.
These perturhations, although small, and, in most in-
stances, insensible in short intervals of time, yet, when
accumulated, as some of them may become, in the lapse
of ages, alter very greatly the original elliptic relations,
so as to render the same elements of the planetary
orbits, which at one epoch represented perfecdy well
their movements, inadequate and unsatisfactory after
long intervals of time.

(490.) When Newton first reasoned his way from
the broad features of the celestial motions, up to the
law of universal gravitation, as aflecting all matter, and
rendering every particle in the universe subject to the
influence of every other, he was not unaware of the
modifications which this generalization would induce
into the results of a more partial and limited application
of the same law to the revolutions of the planets about
the sun, and the satelHtes about their primaries, as their
only centres of attraction. So far from it, that his ex-
traordinary sagacity enaliled him to perceive very dis-
tinctly how several of the most important of the lunar
inequalities take their origin, in this more general way
of conceiving the agency of the attractive power, espe-
cially the retrograde motion of the nodes, and the direct
revolution of the apsides of her orbit. And if he did
not extend his investigations to the mutual perturbations
of the planets, it was not for Avant of perceiving that such
perturbations inust exist, and might go the length of
producing great derangements from the actual state of
the system, but owing to the then undeveloped state of
the practical part of astronomy, which had not yet at-
tained the precision requisite to make such an attempt
inviting, or indeed feasible. What Newton left undone,
however, his successors have accomplished ; and, at
this day, there is not a single perturbation, great or small,

296 A TREATISE ON ASTRONOMY. [ciIAP. XI.

which observation has ever delected, which has not
been traced up to its origin in the mutual gravitation of
the parts of our s)'stein, and been minutely accounted
for, in its numerical amount and value, by strict calcula-
tion on Newton's principles.

(491.) Calculations of this nature require a very high
analysis for their successful performance, such as is far
beyond the scope and object of this work to attempt ex-
hibiting. The reader who would master them must
prepare himself for the undertaking by an extensive
course of preparatory study, and must ascend by steps
which we must not here even digress to point out. It will
be our object, in this chapter, however, to give some
general insight into the nature and manner of operation
of the acting forces, and to point out what are the cir-
cumstances which, in some cases, give them a high de-
gree of efficiency — a sort of purchase on the balance of
the system ; while, in others, with no less amount of
intensity, their effective agency in producing extensive
and lasting changes is compensated or rendered abortive ;
as well as to explain the nature of those admirable re-
sults respecting the stability of our system, to which the
researches of geometers have eonducted them ; and
which, under the form of mathematical theorems of great
beauty, simplicity, and elegance, involve the history of
the past and future state of the planetary orbits during
ages, of which, contemplating the subject in this point
of view, we neither perceive the beginning nor the
end.

(492.) Were there no other bodies in the universe but
the sun and one planet, the latter would describe an
exact ellipse about the former (or both round their com-
mon centres of gravity), and continue to perform its revo-
lutions in one and the same orbit for ever ; but the
moment we add to our combination a third body, the at-
traction of this will draw both the former bodies out of
their mutual orbits, and, by acting on them unequally,
will disturb their relation to each other, and put an end
to the rigorous and mathematical exactness of their ellip-
tic motions, either about one another or about a fixed point
in space. From this way of propounding the subject,
we see that it is not the whole attraction of the newly in-

CHAP. XI.] OF PERTURBATIONS. 297

trodiiced body which produces perturbation, but the dif-
ference of its attractions on the two originally present.

(493.) Compared to the sun, all the planets are of ex-
treme minuteness ; the mass of Jupiter, the greatest of
them all, being not more than one 1300th part that of the
sun. Their attractions on each other, therefore, are all
very feeble, compared with the presiding central power,
and the effects of their disturbing forces are proportionally
minute. In the case of the secondaries, the chief agent
by which their motions are deranged is the sun itself,
whose mass is indeed great, but Avhose disturbing influ-
ence is immensely diminished by their near proximity to
their primaries, compared to their distances from the sun,
which renders the difference of attractions on both ex-
tremely small, compared to the whole amount. In this
case, the greatest part of the sun's attraction, viz. that
which is common to both, is exerted to retain both pri-
mary and secondary in their common orbit about itself,
and prevent their parting company. The small overplus
of force only acts as a disturbing power. The mean
value of this overplus, in the case of the moon disturbed
by the sun, is calculated by Newton to amount to no
higher a fraction than ga^-ro o ¥ ^^ gravity at the earth's
surface, or ^^^ of the principal force which retains the
moon in its orbit.

(494.) From this extreme minuteness of the intensities
of the disturbing, compared to the principal forces, and
the consequent smallness of their momentary effects, it
happens that we can estimate each of these effects sepa-
rately, as if the others did not take place, without fear
of inducing error in our conclusions beyond the limits
necessarily incident to a first approximation. It is a
principle in mechanics, immediately flowing from the
primary relations between forces and the motions they
produce, that when a number of very minute forces act
at once on a system, their joint effect is the sum or ag-
gregate of their separate effects, at least within such limits,
that the original relation of the parts of the system shall
not have been materially changed by their action. Such
effects supervening on the greater movements due to the
action of the primary forces may be compared to the
small rippiings caused by a thousand varying breezes on

298 A TREATISE ON ASTRONOMY. [cHAP. XI.

the broad and regular swell of a deep and rolling ocean,
which run on as if the surface were a plane, and cross in
all directions, without interfering, each as if the other had
no existence. It is only when their eflecls become accu-
mulated in lapse of time, so as to alter the primary rela-
tions or data of the system that it becomes necessary to
have especial regard to the changes correspondingly in-
troduced into the estimation of their momentary efficiency,
by which the rate of the subsequent changes is affected,
and periods or cycles of immense length take their origin.
From this consideration arise some of the most curious
theories of physical astronomy.

(495.) Hence it is evident, that in estimating the dis-
turbing influence of several bodies forming a system, in
which one has a remarkable preponderance over all the
rest, we need not embarrass ourselves with combinations
of the disturbing powers one among another, unless where
immensely long periods are concerned ; such as consist
of many thousands of revolutions of the bodies in ques-
tion about their common centres. So that, in effect, the
problem of the investigation of the perturbations of a
system, however numerous, constituted as ours is, reduces
itself to that of a system of three bodies : a predominant
central body, a disturbing, and a disturbed ; the two lat-
ter of which may exchange denominations, according as
the motions of the one or the other are the subject of
inquiry.

(496.) The intensity of the disturbing force is conti-
nually varying, according to the relative situation of the
disturbing arhd disturbed body with respect to the sun. If
the attraction of the disturbing body M, on the central
body S, and the disturbed body P (by which designa-
tions, for brevity, we shall hereafter indicate them), were
equal, and acted in parallel lines, whatever might other-
wise be its law of variation, there would be no deviation
caused in the elliptic motion of P about S, or of each
about the other. The case would be strictly that of art.
385 ; the attraction of M, so circumstanced, being at
every moment exactly analogous in its effects to terres-
trial gravity, which acts in parallel lines, and is equally
intense on all bodies, great and small. But this is not
the case of nature. Whatever is stated in the subsequent

CHAP. XI.] PROBLEM OF THREE BODIES. 299

article to that last cited, of the disturbing cft'ect of the
sun and moon, is, mutatis mutandis, applicable to every
case of perturbation ; and it must be now our business to
enter, somewhat more in detail, into the general heads of
the subject there merely hinted at.

(497.) We shall begin with that part of the disturbing
force which tends to draAV the disturbed body out of the
plane in which its orbit would be performed if undisturb-
ed, and, by so doing, causes it to describe a curve, of
which no two adjacent portions lie in one plane, or, as it
is called in geometry, a curve of double curvature. Sup-
pose, then, APN to be the orbit wliich P would describe
about S, if undisturbed, and suppose it to arrive at P, at
any instant of time, and to be about to describe in the
next instant the undisturbed arc Vp, which, prolonged in
the direction of its tangent PpR, will intersect the plane
of the orbit ML of the disturbing body, somewhere in the
line of nodes SL, suppose in R. This would be the case
if M exerted no disturbing power. But suppose it to do
so, then, since it draws both S and P towards it, in direc-
tions not coincident with the plane of P's orbit, it will
cause them both, in the next instant of time, to quit that
plane, but iineqiiaUy : — first, because it does not draw
them both in parallel lines ; secondly, because they, being
unequally distant from M, are unequally attracted by it,
by reason of the general law of gravitation. Now, it is
by the difference of the motions thus generated that the
relative orbit of P about S is changed ; so that, if we
continue to refer its motion to S as a fixed centre, the dis-
turbing part of the impulse which it receives from M will
impel it to deviate from the plane PSN, and describe in
the next instant of time, not the arc Pp, but an arc Fq,
lying either above or below P/9, according to the prepon-
derance of the forces exerted by M on P and S.

(498.) The disturbing force acts in the plane of the tri-
angle SPM, and may be considered as resolved into two ;
one of which urges P to or from S, or along the line SP,
and, therefore, increases or diminishes, in so far as it is
effective, the direct attraction of S or P ; the other along
aline PK, parallel to SM, and which may be regarded as
either pulling P in the direction PK, or pushing it in a
contrary direction ; these terms being well understood to

300 A TREATISE ON ASTRONOMY. [cHAP. XI.

have only a relative meaning as referring to a supposed
fixity of S, and transfer of the whole efl'ective power to P.

The former of these forces,
acting always in the plane of
P's motion, cannot tend to
urge it out of that plane : the
latter only is so effective, and
that not wholly ; another reso-
lution of forces being needed
to estimate its efTective part.
But with this we shall not
concern ourselves, the object here proposed being only
to explain the manner in which the motion of the nodes
arises, and not to estimate its amount.

(499.) In the situation, or configuration, as it is termed,
represented in the figure, the force, in the direction PK,
is ^pulling force ; and as PK, being parallel to SM,lies
below the plane of P's orbit (taking that of M's orbit for
a ground plane), it is clear that the disturbed arc Vq, de-
scribed in the next moment by P, must lie beloiv Vp.
When prolonged, therefore, to intersect the plane of M's
orbit, it will meet it in a point r, behind R, and the line
Sr, which will be the line of intersection of the plane
SP</ (now, for an instant, that of P's disturbed motion),
or its new line of nodes, will fall behind SR, the undis-
turbed line of nodes ; that is to say, the line of nodes
will have retrograded by the angle RSr, the motions of
P and M being regarded as direct.

(500.) Suppose, now, M to lie to the left instead of the
right of the line of nodes, P retaining its situation, then
will the disturbing force, in the direction PK, tend to raise
P out of its orbit, to throw P^' above Vp, and r in advance
of R. In this configuration, then, the node will advance ;
but so soon as P passes the node, and comes to the lower
side of M's orbit, although the same disposition of the
forces will subsist, and Vq will, in consequence, continue
to lie above Vp, yet, in this case, the little arc Vq will
have to be prolonged backwards to meet our ground
plane, and, when so prolonged, will lie belo2V the similar
prolongation of Pp, so that, in this case again, the node
will retrograde.

(501.) Thus we see that the effect of the disturbing

CHAP. XI.] MOTION OF THE NODES. 301

force, in the different states of configuration which the
bodies P and M may assume with respect to the node, is
to keep the line of nodes in a continual state of fluctua-
tion to and fro ; and it will depend on the excess of cases
favourable to its advance over those which favour its re-
cess, in an average of all the possible configurations,
whether, on the whole, an advance or recess of the node
shall take place.

(502.) If the orbit of M be very large compared with
that of P, so large tliat MP may, without material error,
be regarded as parallel to MS, which is the case with the
moon's orbit disturbed by the sun, it will be very readily
seen, on an examination of all the possible varieties of
configuration, and having due regard to the direction of
the disturbing force, that during every single complete
revolution of P, the cases favourable to a retrograde mo-
tion of the node preponderate over those of a contrary
tendency, the retrogradation taking place over a- larger
extent of the whole orbit, and being at the same time
more rapid, owing to a more intense and favourable action
of the force than the recess. Hence it follows that, on
the whole, during every revolution of the moon about the
earth, the nodes of her orbit recede on the ecliptic, con-
formable to experience, with a velocity varying from lu-
nation to lunation. The amount of this retrogradation,
when calculated, as it may be, by an exact estimation of
all the acting forces, is found to coincide with perfect
precision with that immediately derived from observation,
so that not a doubt can subsist as to this being the real
process by which so remarkable an effect is produced.

(503.) Theoretically speaking, we cannot estimate
correctly the recess of the intersection of the moon's
orbit with the ecliptic, from a mere consideration of the
disturbance of one of these planes. It is a compound
phenomenon ; both planes are in motion with respect to
an imaginary fixed ecliptic, and, to obtain the compound
effect, we must also regard the earth as disturbed in its
relative orbit about the sun by the moon. But, on ac-
count of the excessive distance of the sun, the intensity
of the moon's attraction on it is quite evanescent, com-
pared with its attraction on the earth : so that the per-
turbative efi'ect in this case, which is the diflference of

2C

302 A TREATISE ON ASTRONOMV. [cHAP. XI.

the moon's attraction on the sun and earth, is equal to
the whole attraction of the moon on the earth. The ef-
fect of this is to produce a monthly displacement of the
centre on either side of the ecliptic, whose amount is
easily calculated by regarding their common centre of
gravity as lying strictly in the ecliptic. From this it ap-
pears, that the displacement in question cannot exceed a
small fraction of the earth's radius in its whole amount ;
and, tlierefore, that its momentary variation, on which the
motion of the node of the ecliptic on the moon's orbit
depends, must be utterly insensible.

(504.) It is otherwise with the mutual action of the
planets. In this case, both the orbits of the disturbed
and disturbing planet must be regarded as in motion.
Precisely on the above-stated principles it maybe shown,
that the effect of each planet's attraction on the orbit of
every other, is to cause a retrogradalion of the node of
the one orbit on the other in certain configurations, and a
recess in others, terminating, like that of the moon, on
the average of many revolutions in a regular retrograda-
tion of the node of each orbit on every other. But since
this is the case with every pair into which the planets can
be combined, the motion ultimately arising from their
joint action on any one orbit, taking into the account the
different situations of all their planes, becomes a singu-
lar and complicated phenomenon, whose law cannot be
very easily expressed in words, though reducible to strict
numerical statement, and being in fact a mere geometri-
cal result of what is above stated.

(505.) The nodes of all the planetary orbits on the^rwe
ecliptic then are retrograde, although (which is a most
material circumstance) they are not all so on a fixed
plane, such as we may conceive to exist in the planetary
system, and to be a plane of reference unaffected by their
mutual disturbances. It is, however, to the ecliptic, that
we are under the necessity of referring their movements
from our station in the system ; and if we would transfer
our ideas to a fixed plane, it becomes necessary to take
account of the variation of the ecliptic itself, produced
by the joint action of all the planets.

(506.) Owing to the smallness of the masses of the
planets, and their great distances from each other, the re-

CHAP. XI.] CHANGE OF INCLINATIONS. 303

volutions of their nodes are excessively slow, being in
every case less tlian a single degree per century, and in
most cases not amounting to half that quantity. So far
as the physical condition of each planet is concerned, it
is evident that the position of their nodes can be of little
importance. It is otherwise with the mutual inclinations
of their orbits, with respect to each other, and to the
equator of each. A variation in the position of tlie eclip-
tic, for instance, by which its pole should shift its dis-
tance from the pole of the equator, would disturb our sea-
sons. Should the plane of the earth's orbit, for instance,
ever be so changed as to bring the ecliptic to coincide
with the equator, we should have perpetual spring over
all the world ; and, on the other hand, should it coincide
with a meridian, the extremes of summer and winter
would become intolerable. The inquiry, then, of the
variations of inclination of the planetary orbits inter se,
is one of much higher practical interest than those of
their nodes.

(507.) Referring to the figure of art. 498, it is evident
that the plane SP^, in which the disturbed body moves
during an instant of time from its quitting P, is diflerently
inclined to the orbit of M, or to a fixed plane, from the
original or undisturbed plane PSp. The difference of
absolute position of these two planes in space is the an-
gle made between the planes PSR and PSr, and is there-
fore calculable by spherical trigonometry, when the angle
RSr or the momentary recess of the node is known, and
also the inclination of tlie planes of the orbits to each
other. We perceive, then, that between the momentary
change of inclination and the momentary recess of the
node there exists an intimate relation, and that the re-
search of the one is in fact bound up in that of the other.
This may be, perhaps, made clearer, by considering the
orbit of M to be not merely an imaginary line, but an
actual circular or elliptic hoop of some rigid material,
without inertia, on which, as on a wire, the body P may
slide as a bead. It is evident that the position of this
hoop will be determined at any instant, by its inclination
to the ground plane to which it is referred, and by the
place of its intersection therewith, or node. It will also
be determined by the momentary direction of P's motion,

304 A TREATISE ON ASTRONOMY. [cHAP. XI.

which (having no inertia) it must obey ; and any change
by which P should, in the next instant, alter its orbit,
would be equivalent to a shifting, bodily, of the whole
hoop, changing at once its inclination and nodes.

(508,) One immediate conclusion from what has been
pointed out above, is that where the orbits, as in the case
of the planetary system and the moon, are slightly in-
clined to one another, the momentary variations of the
inclination are of an order much inferior in magnitude to
those in the place of the node. This is evident on a
mere inspection of our figure, the angle RPr, being by
reason of the small inclination of the planes SPR and
RSr, necessarily much smaller than the angle HSr. In
proportion as the planes of the orbits are brought to coin-
cidence, a very trilling angular movement of Pp about PS
as an axis will make a great variation in the situation of the
point r, where its prolongation intersects the ground plane.

(509.) To pass from the momentary changes which
take place in the relations of nature to the accumulated
eftects produced in considerable lapses of time by the
continued action of the same causes, under circumstances
varied by these very effects, is the business of the integral
calculus. Without going into any calculations, however,
it will be easy for us to trace, by a few cases, the varying
influence of differences of position of the disturbing and
disturbed body with respect to each other and to the node,
and from these to demonstrate the two leading features
in this theory — the periodic nature of the change and
re-establishment of the original inclinations, and the
small limits within which these changes are confined.

(510.) Case 1. — When the di.sturl)ing body M is situ-
ated in a direction perpendicular to the line of nodes, or

CHAP,

XI.]

MOTION OF THE NODES.

305

the nodes are in quadrature with it : M being the dis-
turbing body, and SN the line of nodes, the disturbing
force will act at P, in the direction PK ; being a pulling
force when P is in any part of the semicircle HAN, and
a pushing force in the whole of the opposite semicircle.
And it is easily seen that this force is greatest at A and
B, and evanescent at H and N. Hence, in the whole
semicircle HAftfP(7 will lie below Vp, and being pro-
duced backwards in the quadrant HA, and forvvards in
AN, will meet the circle S6N« in the plane of M's
orbit, in points behind the nodes SN, the nodes being
retrograde in both cases. But the new inclination of
the disturbed orbit is, in the former case, PxA, Avhich
"is less than PHa ; and in the latter, Vya, which is
greater than PN«. In the other semicircle the direction
of the disturbing force is changed ; but that of the motion,
with respect to the plane of M's orbit, being also in
each quadrant reversed, the same variations of node and
inclination will be caused. In this situation of M, then,
the nodes recede during every part of the revolution of
P, but the inclination diminishes throughout the quadrant
$A, increases again by the same identical degrees in the
quadrant AN, decreases throughout the quadrant N6,
and is finally restored to its pristine value at SL^i Oi^ the
average of a revolution of P, supposing M unmoved, the
nodes will have retrograded with their utmost speed, but
the inclination Avill remain unaltered.

(511.) Case 2. — Suppose the disturbing body now to
be fixed in the line of nodes, or the nodes to be in
syzygy, as in the annexed figure. In this situation the
direction of the disturbing force, which is always parallel
to SM, lies constantly in the plane of P's orbit, and there-

ac3

306 A TREATISE ON ASTRONOMY, [cHAP. XI.

fore produces neither variation of inclination or motion
of nodes.

(512.) Case 3. — Let us take now an intermediate
situation of M, and indicating by the arrows the directions
of the disturbing forces (which are pulUng ones through-
out all the semi-orbit which lies towards M, and pushing
in the opposite), it will readily appear that the reasoning
of art. 510, will hold good in all that part of the orbit
which lies between T and N, and between V and H,
but that the effect will be reversed by the reversal of the
direction of the motion with respect to the plane of M's
orbit, in the intervals HT and NV. In these portions,
hoAvever, the disturbing force is feebler than in the others,
being evanescent in the line of q^iadratures TV, and in-

T

creasing to its maximum in the syzygics a h. The nodes
then will recede rapidly in the former intervals, and ad-
vance feebly in tlie latter ; but since, as II approaches to
a, the disturbing force, by acting obliquely to the plane
of P's orbit, is again diminished in efficacy, still, on the
average of a whole revolution, the nodes recede. On
the other hand, the inclination will now diminish during
the motion of P from T to c, a point 90° distant from
the node, while it increases not only during its whole
motion over the quadrant cN, but also in the rest of its
half revolution NV, and so for the other half. There
will, therefore, be an uncompensated increase of inclina-
tion in this position of M, on the average of a whole
revolution.

(513.) But this increase is converted into diminution
when the line of nodes stands on the other side of SM,
or in the quadrants Yb, Ta ; and still regarding M as
fixed, and supposing that the change of circumstances

CHAP. XI.3 CHANGES OF INCLINATION. 307

arises not from the motion of M but iVoni that of the
node, it is evident that so soon as tlic line of nodes in
its retrograde motion has got past a, tlie circumstances
will be all exactly reversed, and the inclination will again
be augmented in each revolution by the very same steps
taken in reverse order by which it before diminished.
On the average, therefore, of a whole revolution of
THE NODE, the inclination will be restored to its original
state. In fact, so far as the mean or average effect on
the inclination is concerned, instead of supposing M
fixed in one position, we might conceive it at every in-
stant divided into four equal parts, and placed at equal
angles on either side of the line of nodes, in which case
it is evident that the effect of two of the parts would be
to precisely annihilate that of the others in each revo-
lution of P.

(514.) In what is said, we have supposed M at rest;
but the same conclusion, as to the mean and final results,
holds good if it be supposed in motion ; for in the
course of a revolution of the nodes, which, owing to the
extreme smallness of their motion, in the case of the
planets, is of immense length, amounting, in most cases,
to several hundred centuries, and in that of the moon
is not less than 237 lunations, the disturbing body M
is presented by its own motion, over and over again, in
every variety of situation to the line of nodes. Before
the node can have materially changed its position, M has
performed a complete revolution, and is restored to its
place ; so that, in fact (that small difference excepted
which arises from the recess of the node in one syno-
dical revolution of M), we may regard it as occupying at
every instant every point of its orbit, or rather as having
its mass distributed uniformly like a solid ring over its
whole circumference. Thus the compensation which
we have shown would take place in a whole revolution
of the node, does, in fact, take place in every synodic
period of M, tliat minute difference only excepted which
is due to the cause just mentioned. This difference,
then, and not the whole disturbing eff'ect of M, is what
produces the effective variation of the inclinations, whe-
ther of the lunar or planetary orbits ; and this difference,
which remains uncompensated by the motion of M, is in

308 A TREATISE ON ASTRONOMY. [cHAP. XI.

its turn rompensated by the motion of the node during
its whole revolution.

(51.^.) It is clear, therefore, that the total variation of
the planetary inclinations must be comprised within very
narrow limits indeed. Geometers have accordingly de-
monstrated, by an accurate analysis of all the circum-
stances, and an exact estimation of tlie acting forces,
that such is the case ; and this is what is meant by as-
serting the stability of the planetary system as to the
mutual inclinations of its orbits. By the researches of
Lagrange (of whose analytical conduct it is impossible
here to give any idea), the following elegant theorem has
been demonstrated: —

" If the mass of every planet be multiplied by the
square root of the major axis of its orbit, and the pro-
duct by the Square of the tangent of its inclination to a
fixed plane, the sum of all these products will be con-
stantly the same under the influence of their mutual at-
traction.^^ If the present situation of tlie plane of the
ecliptic be taken for that fixed plane (the ecliptic itself
being variable like the other orbits), it is found that this
sum is actually very small ; it must, therefore, always
remain so. This remarkable theorem alone, then, would
guarantee the stability of tlie orbits of the greater planets ;
but from what has above been shown, of the tendency of
each planet to work out a compensation on every other,
it is evident that the minor ones are not excluded from
this beneficial arrangement.

(516.) Meanwhile, there is no doubt that the plane
of the ecliptic does actually vary by the actions of the
planets. The amount of this variation is about 48" per
century, and has long been recognised by astronomers,
by an increase of the latitudes of all the stars in certain
situations, and their diminution in the opposite regions.
Its effect is to bring the ecliptic by so much per annum
nearer to coincidence with the equator ; but from what
Ave have above seen, this diminution of the obliquity of
the ecliptic will not go on beyond certain very moderate
limits, after which (although in an immense period of
ages, being a compound cycle resulting from the joint
action of all the planets) it will again increase, and thus
oscillate backward and forward about a mean position,

CHAP. XI.] PRKCESSION* OF THE EQUINOXES. 309

the extent of its deviation to one side and the other being
less than 1° 21'.

(517.) One effect of this variation of the plane of the
ecliptic, that which causes its nodes on a fixed plane
to change — is mixed up with the precession of the
equinoxes (art. 261), and undislinguishable from -it, ex-
cept in theory. This last-mentioned phenomenon is,
however, due to another cause, analogous, it is true, in a
general point of view to those above considered, but
singularly modified by the circumstances under which it
is produced. We shall endeavour to render these modi-
fications intelligible, as far as they can be made so, with-
out the intervention of analytical formulae.

(518.) The precession of the e([uinoxes, as we have
shown in art. 2(50, consists in a continual retrograda-
tion of the node of the earth's equator on the ecliptic,
and is, therefore, oliviously an effect so far analogous to
the general phenomenon of the retrogradation of the
nodes of the orbits on each other. The immense dis-
tance of the planets, however, compared with the size
of the earth, and the smallness of their masses com-
pared to that of the sun, puts tlieir action out of the
question in the inquiry of its cause, and we must,
therefore, look to tlie massive though distant sun, and
to our near though minute neighbour, the moon, for its
explanation. This will, accordingly, be found in their
disturl)inff action on the redundant matter accumulated
on the equator of the earth, by which its figure is ren-
dereil spheroidal, combined with the eartli's rotation on
its axis. It is to the sagacity of Newton that we owe
the discovery of this singular mode of action.

(519.) Suppose in our figures (arts. 509, 510, 511),
that instead of one body, P, revolving round S, there
were a succession of particles not coherent, but forming
a kind of fluid ring, free to change its forni by any force
applied. Then, while this ring revolved round S in its
own plane, under the disturbing influence of the distant
body M (which now represents the moon or the sun,
as P does one of the particles of the earth's equator),
two things would happen: — 1st, Its figure would be
bent out of a plane into an undulated form, those parts
of it within the arcs Vc and T(/ {fig. art. 511) being

310 A TREATISE ON ASTRONOMY. [cHAP. XI.

rendered more inclined to the plane of M's orbit, and
those within the arcs cT, f/V, less so that they would
otherwise be. 2dly, the nodes of this ring, regarded as
a Avhole, without respect to its change of figure, would
retreat upon that plane.

(520.) But suppose this ring, instead of consisting
of discrete molecules free to move independently, to be
rigid and incapable of such flexure, like the hoop we
have supposed in art. 507, then it is evident that the
effort of those parts of it which tend to become more
inclined will act through the medium of the rins: itself
(as a mechanical engiiie or lever) to counteract the
effort of those which have at the same instant a contrary
tendency. In so far only, then, as there exists an excess
on the one or the other side will the inclination change,
an average being struck at every moment of the ring's
motion ; just as was shown to happen in the view we
have taken of the inclinations, in every complete revolu-
tion of a single disturbed body, under the influence of a
fixed disturbing one.

(521.) Meanwhile, however, the nodes of the rigid
ring will retrograde, \\\e general or average tendency of
the nodes of every molecule being to do so. Here, as
in the other case, a struggle will take place by the coun-
teracting efforts of the molecules contrarily disposed,
propagated through the solid substance of the ring ; and
thus, at every instant of time, an average will be struck,
which average being identical in its nature with that ef-
fected in the complete revolution of a single disturbed
body, will, in every case, be in favour of a recess of the
node, save only when the disturbing body, be it sun or
moon, is situated in the plane of the earth's equator, or
in the case of the Jig. art. 510. '

(522.) This reasoning is evidently independent of any
consideration of the cause which maintains the rotation
of the ring ; whether the particles be small satellites re-
tained in circular orbits under the equilibrated action of
attractive and centrifugal forces, or whether they be small
masses conceived as attached to a set of imaginary spokes
as of a wheel, centering in S, and free only to shift their
planes by a motion of those spokes perpendicular to the
plane of the wheel. This makes no difference in the

CHAP. XI.3 PRECESSION OF THE EQUINOXES. 311

general effect ; though the different velocities of rotation,
which may be impressed on such a system, may and
will have a very great iniluence both on the absolute and
relative magnitudes of the two effects in question — the
motion of the nodes and change of inclination. This
M'ill be easily understood, if we suppose the ring without
a rotatory motion, in which extreme case it is obvious,
that so long as M remained fixed there would take place
no recess of nodes at all, but only a tendency of the ring
to tilt its plane round a diameter perpendicular to the
position of M, bringing it towards the line SM.

(523.) The motion of such a ring, then, as we have
been considering, would imitate, so far as the recess of
the nodes goes, the precession of the equinoxes, only that
its nodes would retrograde far more rapidly than the ob-
served precession, which is excessively slow. But now
conceive this ring to be loaded with a spherical mass
enormously heavier than itself, placed concentrically
within it, and cohering firmly to it, but indifferent, or very
nearly so, to any such cause of motion ; and suppose,
moreover, that instead of one such ring, there are a vast
multitude heaped together around the equator of such a
globe, so as to form an elliptical protuberance, enveloping
it like a shell on all sides, but whose mass, taken together,
should form but a very minute fraction of the whole
spheroid. We have now before us a tolerable repre-
sentation of case of nature ;* and it is evident that the
rings, having to drag round with them in their nodal re-
volution this great inert mass, will have their velocity of
retrogradation proportionally diminished. Thus, then, it
is easy to conceive how a motion, similar to the preces-

* That a perfect sphere would be so inert and indifferent as to a revo-
lution of the nodes of its equator under the influence of a distant attract-
ing body appears from this — that the direction of the resultant attraction
of such a body, or of that single force which, opposed, would neutralize
and destroy its whole action, is necessarily in a line passing through the
centre of the sphere, and, therefore, can have no tendency to turn the
sphere one way or other. It may be objected by the reader, that the
whole sphere may be conceived as consisting of rings parallel to its
equator, of every possible diameter, and that, therefore, its nodes should
retrograde even without a protuberant equator. The inference is in-
correct, but our limits will not allow us to go into an exposition of the
fallacy. We should, however, caution him, generally, that no dynamical
subject is open to more mistakes of this kind, which notliing but the
closest attention, iii every varied point of view, will detect.

312 A TREATISE ON ASTRONOMY. [cHAP. XI.

sion of the equinoxes, and, like it, characterized by ex-
treme slowness, will arise from the causes in action.

(524.) Now a recess of the node of the earth's equa-
tor, upon a given plane, corresponds to a conical motion
of its axis round a perpendicular to that plane. But in the
case before us, that plane is not the ecliptic, but the moon's
orbit for the time being ; and it may be asked how we
are to reconcile this with wdiat is stated in art. 266, re-
specting the nature of the motion in question. To this
Ave reply, that the nodes of the lunar orbit, being in a state
of continual and rapid retrogradation, while its inclination
is preserved nearly invariable, the point in the sphere of
the heavens rouncl which the pole of the earth's axis re-
volves (with that extreme slowness characteristic of the
precession) is itself in a state of continual circulation
round the pole of the ecliptic, with that much more rapid

motion which belongs to the lunar
node. A glance at the annexed
figure will explain this better than
words. P is the pole of the eclip-
tic, A the pole of the moon's orbit,
moving round tlie small circle
ABCD in 19 years ; a the pole of
the earth's equator, which at each
moment of its progress has a direc-
tion perpendicular to the varying
position of the line Ao, and a velo-
city depending on the varying in-
tensity of the acting causes during
the period of the nodes. This ve-
locity, however, being extremely small, when A comes
to B, C, D, E, the line A« will have taken up the positions
B6, Cc, \)d, Ee, and the earth's pole a will thus, in one
tropical revolution of the node, have arrived at e, having
described not an exactly circular arc, but a single undu-
lation of a wave-shaped or epicycloidal curve, ab ode,
with a velocity alternately greater and less than its mean
motion, and this will be repeated in every succeeding
revolution of the node.

(525.) Now this is precisely the kind of motion which,
as we have seen in art. 272, the pole of the earth's equa-
tor really has round the pole of the ecliptic, in conse

■V

CHAP. XI.] NUTATION. 313

quence of the joint effects of precession and nutation,
which are thus uranographically represented. If we
superadd to the elfeet of hinar precession that of the so-
lar, which alone would cause the pole to describe a circle
uniformly about P, this wjll only affect the undulations
of our waved curve, by extending tliem in length, but
will pi'odiice no effect on the depth of the waves, or the
excursions of tlie earth's axis to and from the pole of the
ecliptic. Thus we see that the two phenomena of nu-
tation and precession are intimately connected, or rather,
both of them essential constituent parts of one and the
same phenomenon. It is hardly necessary to state that
a rigorous analysis of this great problem, by an exact es-
timation of all the acting forces and summation of their
dynamical effects,* leads to the precise value of the co-
efficients of precession and nutation, which observation
assigns to them. The solar and lunar portions of the
precession of the equinoxes, that is to say, those portions
Avhich are uniform, are to each other in the proportion
of about 2 to 5.

(526.) In the nutation of the earth's axis we have an
example (the. first of its kind which has occurred to us)
of a periodical movement in one part of the system,
giving rise to a motion having the same pi'ecise period
in another. The motion of the moon's nodes is here,
we see, represented, though under a very different form,
yet in the same exact periodic time, by the movement
of a peculiar oscillatory kind impressed on the solid
mass of the earth. We must not let the opportunity pass
of generalizing the principle involved in this result, as it
is one which we shall find again and again exemplified in
every part of physical astronomy, nay, in every depart-
ment of natural science. It may be stated as "the prin-
ciple of forced oscillations, or of forced vibrations," and
tlius generally announced : —

If one pari of any system connected either by 7nafe-
rial ties, or by the mutual attractions of its members.,
he continually maintained by any cause, whether in-
herent in the constitution of the system or external to
it, in a state of regular periodic motion, that motion
will be propagated throughout the ivhole system, and

* Vide Prof. Airy's Mathematical Tracts, 2d cd. p 200, &c.
2D

314 A TREATISE ON ASTRONOMY. [cHAP. XI.

will give rise in every member of it, and in every -part
of each member, to periodic movements executed in
equal periods with that to which they owe their origin,
though not necessarily synchronous ivith them in their
maxima and minima*

The system may be favourably or unfavourably con-
stituted for such a transfer of periodic movements, or
favourably in some of its parts and unfavourably in
others ; and, accordingly as it is the one or the other,
the derivative oscillation (as it may be termed) will be
imperceptible in one case, of appreciable magnitude in
another, and even more perceptible in its visible effects
than the original cause, in a tliird; of this last kind we
have an instance in the moon's acceleration to be here-
after noticed.

(527.) It so happens that our situation on the earth,
and the delicacy which our observations have attained,
enable us to make it, as it were, an instrument to feel these
forced vibrations — these derivative motions, communi-
cated from various quarters, especially from our near
neighbour, the moon, much in the same way as we de-
tect, by the trembling of a board beneath us, the secret
transfer of motion by which the sound of an organ pipe
is dispersed through the air, and carried down into the
earth. Accordingly, the monthly revolution of the moon,
and the annual motion of the sun, produce, each of them,
small nutations in the earth's axis, whose periods are
respectively half a month and half a year, each of which,
in this view of the subject, is to be regarded as one por-
tion of a period consisting of two equal and similar parts.
But the most remarkable instance, by far, of this propa-
gation of periods, and one of high importance to man-
kind, is that of the tides, which are forced oscillations,
excited by the rotation of the earth in an ocean disturbed
from its figure by the varying attractions of the sun and
moon, each revolving in its own orbit, and propagating
its own period into the joint phenomenon.

(528.) The tides are a subject on which many persons
find a strange diificulty of conception. That the moon, by

* See a demoastration of (his theorem for the forced vibrations of sys-
tems comiected by material ties of imperfect elasticity, in my treatise on
Sound, Eneyc. Metrop. art. 323. The demonstration is easily extended
and generalized to take in other systems. — Author.

CHAP. XI.] THE TIDES. 315

her attraction, should heap up the \vaters of the ocean
under her, seems to most persons very natural — that
the same cause should, at the same time, heap them up
on the opposite side, seems to many palpably absurd.
Yet nothing is more true, nor indeed more evident, when
we consider that it is not by her ivhole attraction, but by
the differences of her attractions at the two surfaces and
at the centre that the waters are raised — that is to say,
by forces directed precisely as the arrows in our figure,
art. 510, in which we may suppose M the moon, and P
a particle of water on the earth's surface. A drop of
water existing alone would take a spherical form, by
reason of the attraction of its parts ; and if the same
drop were to fall freely in a vacuum under the influence
of an uniform gravity, since every part would be equally
accelerated, the particles would retain their relative posi-
tions, and the spherical form be unchanged. But sup-
pose it to fall under the influence of an attraction acting
on each of its particles independently, and increasing
in intensity at every step of the descent, then the parts
nearer the centre of attraction would be attracted more
than the central, and the central than the more remote,
and the whole would be drawn out in the direction of the
motion into an oblong form ; the tendency to separation
being, however, counteracted by the attraction of the
particles on each other, and a form of equilibrium being
thus established. Now, in fact, the earth is constantly
falling to the moon, being continually drawn by it out
of its path, the nearer parts more and the remoter less
so than the central ; and thus, at every instant, the moon's
attraction acts to force down the water at the sides, at
right angles to her direction, and raise it at the two ends
of the diameter pointing towards her. Geometry corro-
borates this view of the subject, and demonstrates that
the form of equilibrium assumed by a layer of water
covering a sphere, under the influence of the moon's at-
traction, would be an oblong ellipsoid, having the semi-
axis directed towards the moon longer by about 58 inches
than that transverse to it.

(529.) There is never time, however, for this spheroid
to be fully formed. Before the waters can take their
level, the moon has advanced in her orbit, both diurnal

316 A TRHATISE ON ASTRONOMY. [cHAP. XI.

and monthly (for in this theory it will answer the pur-
pose of clearness better if we suppose the earth's diurnal
motion transferred to the sun and moon in the contrary
direction), the vertex of the spheroid has shifted on the
earth's surface, and the ocean has to seek a new bearing.
The effect is to produce an immensely broad and exces-
sively flat wave (not a circulating current), which follows,
or endeavours to follow, the apparent motions of the
moon, and must, in fact, if the principle of forced vibra-
tions be true, imitate by equal, though not by synchro-
nous, periods, all the periodical inequalities of that motion.
When the higher or lower parts of this wave strike our
coasts, they experience what we call high and low water.

(530.) The sun also produces precisely such a wave,
whose vertex tends to follow the apparent motion of the
sun in the heavens, and also to imitate its periodic in-
equalities. This solar wave coexists with the lunar —
is sometimes superposed on it, sometimes transverse to it,
so as to partly neutralize it, according to the monthly
synodical configuration of the two luminaries. This al-
ternate mutual reinforcement and destruction of the solar
and lunar tides cause what are called the spring and
neap tides — the former being their sum, the latter their
difference. Although the real amount of either tide is,
at present, hardly within the reach of exact calculation,
yet their proportion at any one place is probably not
very remote from that of the ellipticities which would
belong to their i-espective spheroids, could an equilibrium
be attained. Now these ellipticities, for the solar and
lunar spheroids, are respectively about two and five feet;
so that the average spring tide will be to the neap as 7
to 3, or thereabouts.

(531.) Another effect of the combination of the solar
and lunar tides is what is called ih.e priming and lagging
of the tides. If the moon alone existed, and moved in
the plane of the equator, the tide-day (i. e. the interval
between two successive arrivals at the same place of the
same vertex of the tide-wave) would be the lunar day
(art. 115) formed by the combination of the moon's si-
dereal period and that of the earth's diurnal motion.
Similarly, did the sun alone exist, and move always on
the equator, the tide-day would be the mean solar day.

CHAP. XI.] THE TIDES. 317

The actual tide-day, then, or the interval of the occur-
rence of two successive maxima of their superposed
waves, will vary as the separate waves approach to or
recede from coincidence ; becEruse, when the vei'tices of
two waves do not coincide, their joint height has its
maximum at a point intermediate between them. This
variation from uniformity in the lengths of successive
tide-days is particularly to be remarked about the time
of the new and full moon.

(.'5.32.) Quite different in its origin is that deviation of
tlie time of high and low water at any port or harbour,
from the culmination of the luminaries, or of the theo-
retical maximum of their superposed spheroids, which
is called the " esta1)lishment" of that port. If the water
were without inertia, and free from obstruction, either
owinof to the friction of the bed of the sea — the narrow-
ness of channels along which the wave has to travel be-
fore reaching the port — their length, &c. &c., the times
above distinguished would be identical. But all these
causes tend to create a difference, and to make that dif-
ference not alike at all ports. The observation of the
establishment of harbours is a point of great maritime
importance ; nor is it of less consequence, theoretically
speaking, to a knowledge of the true distribution of the
tide waves over the globe.* In making such observa-
tions, care must be taken not to confound the time of
" slack water," when the current caused by the tide ceases
to flow visibly one way or the other, and that of high or loiv
water, when the level of the surface ceases to rise or fall.
These are totally distinct phenomena, and depend on en-
tirely different causes, though it is true they may some-
times coincide in point of time. They are, it is feared,
too often mistaken one for the other by practical men ; a
circumstance which, whenever it occurs, must produce
the greatest confusion in any attempt to reduce the sys-
tem of the tides to distinct and intelligible laws.

(533.) The declination of the sun and moon materially

* The recent investigations of Mr. Lubbock, and those highly interest-
ing ones in whicli Mr. Whevvell is understood to be engaged, will, it is
to be hoped, not only throw theoretical light on the very otecure sub-
ject of the tides, but (what is at present quite as much wanted) arouse
the attention of observers, and at the same time give it that right direc-
tion, by pointing out wliai ought to be observed, without v^hich all obser-
vation is lost labour.

2d2

318 A TREATISE ON ASTRONOMY. [cHAP. XI.

affects the tides at any particular spot. As the vertex of
the tide-wave tends to place itself vertically under the
luminary which produces it, when this vertical changes
its point of incidence on the surface, the tide-wave must
tend to shift accordingly, and thus, by monthly and an-
nual periods, must tend to increase and diminish alter-
nately the principal tides. The period of the moon's
nodes is thus introduced into this subject; her excursions
in declination in one part of that period being 29°, and
in another only 17°, on eitlier side the equator.

(534.) Geometry demonstrates that the efficacy of a
luminary in raising tides is inversely proportional to the
cube of its distance. The sun and moon, however, by
reason of the ellipticity of their orbits, are alternately
nearer to and fartlier from the earth than their mean dis-
tances. In consequence of this, the efficacy of the sun
will fluctuate between the extremes 19 and 21, taking
20 for its mean value, and that of the moon between 43
and 59. Taking into account this cause of difference,
the highest spring tide will be to the lowest neap as 59
+21 to 43 — 19, or as 80 to 24, or 10 to 3. Of all the
causes of differences in the height of tides, however,
local situation is the most influential. In some places,
the tide-wave, rusliing up a narrow channel, is suddenly
raised to an extraordinary height. At Annapolis, for
instance, in the Bay of Fmidy, it is said to raise 120
feet.* Even at Bristol, the difference of high and low
water occasionally amoimts to 50 feet.

(535.) The action of the sun and moon, in like man
ner, produces tides in the atmosphere, which delicate
observations have been able to render sensible and mea-
surable. This effect, however, is extremely minute.

(536.) To return, now, to tlie planetary perturbations.
Let us next consider the changes induced by their mu-
tual action on the magnitudes and forms of their orbits,
and in their positions therein in different situations with
respect to each other. In the first place, however, it
will be proper to explain the conventions under which
geometers and astronomers have alike agreed to use the
language and laws of the elliptic system, and to continue
to apply them to disturbed orbits, although those orbits
* Robison's Lectures on Mechanical Philosophy.

CHAP. XI.] THE TIDES. 319

SO disturbed arc no longer, in mathematical strictness,
ellipses, or any known curves. This they do, partly on
account of the convenience of conception and calcula-
tion vvhicli attaches to this system, but much more for
this reason — that it is found, and may be demonstrated
from tlie dynamical relations of the case, that the de-
parture of each planet from its ellipse, as determined at
any epoch, is capable of being truly represented, by sup-
posing the ellipse itself to be slowly variable, to change
its matrnitude and eccentricity, and to shift its position
and the plane in which it lies according to certain laws,
while the planet all the time continues to move in this
ellipse, just as it would do if the ellipse remained in-
variable and the distar])ing forces had no existence. By
this way of considering the subject, the whole permanent
effect of the disturbing forces is regarded as thrown upon
the orbit, while the relations of the planet to that orbit
remain unchanged, or only liable to brief and compara-
tively momentary ilnctuation. This course of procedure,
indeed, is the most natural, and is in some sort forced upon
us by the extreme slowness with which the variations
of the elements develope themselves. For instance, the
fraction expressing the eccentricity of the earth's orbit
changes no moi'e than 0-00004 in its amount in a cen-
tury ; and the place of its perihelion, as referred to the
sphere of the heavens, by only 19' 39" in the same
time. For several years, therefore, it would be next to
impossible to distinguish between an ellipse so varied
and one that had not varied at all ; and in a single revo-
lution, the difi'erence between the original ellipse and
the curve really represented by the varying one, is so
excessively minute, that if accurately drawn on a table,
six feet in diameter, the nicest examination Avith mi-
croscopes, continued along the whole outlines of the two
curves, would hardly detect any perceptible interval be-
tween them. Not to call a motion so minutely conform-
ing itself to an elliptic curve, elliptic, would be affecta-
tion, even granting the existence of trivial departures
alternately on one side or on the other ; though, on the
other hand, to neglect a variation, which continues to
accumulate from age to age, till it forces itself on our
notice, would be wilful blindness.

320 A TREATISE ON ASTRONOMY. [cHAP. XI.

(537.) Geometers, then, have agreed in each single
revohition, or for any ninderate interval of time, to re-
gard the motion of each planet as elliptic, and performed
according to Kepler's laws, willi a reserve in favour of
certain very small and transient fluctuations, but at the
same time to regard all the elements of each ellipse as
in a continual, though extremely slow, state of change ;
and, in tracing the effects of perturbation on the system,
they take account principally, or entirely, of this change
of the elements, as that upon which, after all, any mate-
rial change in the great features of the system will ulti-
mately depend.

(538.) And here we encounter the distinction between
what are termed secular variations, and such as are ra-
pidly periodic, and are compensated in short intervals.
In our exposition of the variation of the inclination of a
disturbed orbit (art. 514), for instance, we showed that,
in each single revolution of the disturbed body, the plane
of its motion underwent fluctuations to and fro in its
inclination to that of the disturbing body, which nearly
compensated each other; leaving, however, a portion
outstanding, which again is nearly compensated by the
revolution of the disturbing body, yet still leaving out-
standing and uncompensated a minute portion of the
change, which requires a whole revolution of the node
to compensate and bring it back to an average or mean
value. Now, the two lirst compensations which are
operated by the planets going through the succession of
configurations with each other, and therefore in compa-
ratively short periods, are called periodic variations ;
and the deviations thus compensated are called inequa-
lities depending on conjigu rations ; while the last,
which is operated by a period of the node (one of the
elements'), has nothing to do with the configurations of
the individual planets, requires an immense period of
time for its consummation, and is, therefore, distinguish-
ed from the former by the term secular variation.

(539.) It is true, that, to afford an exact representation
of the motions of a disturbed body, whether planet or
satellite, both periodical and secular variations, Avith
their corresponding inequalities, require to be express-
ed; and, indeed, the former even more than tlie latter;

CHAP. XI.3 VARIATIONS, PERIODIC AND SECULAR. 321

seeing that the secular inequalities are, in fact, nothing
but what remains after the mutual destruction of a much
larger amount (as it very oi'tcn is) of periodical. But
these are in their nature transient and temporary : they
disappear, and leave no trace. The planet is tempora-
rily drawn from its orbit (its slowly varying orbit), but
forthwith returns to it, to deviate presently as much the
other way, while the varied orbit accomodates and ad-
justs itself to the average of these excursions on either
side of it; and thus continues to present, for a succes-
sion of indefinite ages, a kind of medium picture of all
that the planet has been doing in their lapse, in which
the expression and character is preserved; but the in-
dividual features are merged and lost. These periodic
inequalities, however, are, as we have observed, by no
means to be neglected, but they are taken account of by
a separate process, independent of the secular variations
of the elements.

(540.) In order to avoid complication, while endea-
vouring to give the reader an insight into both kinds of
variations, we shall henceforward conceive all the orbits
to lie in one plane, and confine our attention to the case
of two only, that of the disturbed and disturbing body,
a view of the subject which (as we have seen) compre-
hends the case of the moon disturbed by the sun, since
any one of the bodies may be regarded as fixed at plea-
sure, provided we conceive all its motions transferred in

a contrary direction to each of the others. Suppose,
therefore, S to be the central, M the disturbing, and P
the disturbed body. Then the attraction of M acts on
P in the direction PM, and on S in the direction SM
And the disturbing part of M's attraction, being the dif-
ference only of these forces, will have no fixed direction,

322 A TREATISE ON ASTRONOMY. j^CHAP. XI.

but will act on P very di (Trrently, accordino- to the configu-
rations of P and M. It will therefore be necessary, in
analyzing its effect, to resolve it, according to niecliani-
cal principles, into forces acting according to some cer-
tain directions ; viz. along the radius vector SP, and per-
pendicular to it. The simplest way to do this, is to resolve
the attractions of M on both S and P in these directions,
and take, in both cases, their difference, which is tlie dis-
turbing part of M's effect. In this estimation, it will be
found then that two distinct disturbing powers originate ;
one, which we shall call the tangential force, acting in
the direction PQ, perpendicular to SP, and therefore in
that of a tangent to the orbit of P, supposed nearly a cir-
cle— the other, which may be called the radial disturl)ing
force, whose direction is always either to or from S.

(541.) It is the former alone (art. 419) which disturbs
the equable description of areas of P about S, and is
therefore tlie chief cause of its angular deviations from
the elliptic place. For the equable description of areas
depends on no particular law of centi'al force, but only
requires that the acting force, whatever it be, should be
directed to the centre ; whatever force does not conform
to this condition, must disturb the areas.

(542.) On the other hand, the radial portion of the dis-
turbing force, though, being always directed to or from
the centre, it does not affect the equable description of
areas, yet, as it does not conform in its law of variation
to that simple law of gravity liy which the elliptic figure
of the orbits is produced and maintained, has a tendency
to disturb this form ; and, causing the disturbed body P,
now to approach the centre nearer, now to recede iiirther
from it, than the laws of elliptic motion would warrant, and
to have its points of nearest approach and farthest recess
otherwise situated than they would be in the undisturbed
orbit, tends to derange the magnitude, eccentricity, and
position of the axis of P's ellipse.

(543.) If we consider the variation of the tangential
force in the different relative positions of M and P, we
shall find that, generally speaking, it vanishes when P is
at A or C, see Jig. to art. 540, i. e. in conjunction with
M, and also at two points, B and D, where JVl is equi-
distant from S and P (or very nearly in the quadratures of

CHAP. XI.] EFFKCTS OF THE TANGENTIAL FORCE. 823

P with M) ; and that, between A and B, or D, it tends to
urge P towards A, while, in the rest of the orbit, its
tendency is to urge it towards C. Consequently, the
general elfect will be, that in P's progress through a com-
plete synodical i-evolution round its orbit from A, it will
first be ac"9^1erated from A up to Br— thence retarded till
it arrives at C — thence again accelerated up to D, and
again retarded till its re-arrival at the conjunction A.

(544.) If P's orbit were an exact circle, as well as M's,
it is evident tliat the retardation which takes place during
the description of the arc AB vv^ould be exactly compen-
sated by the acceleration in the arc DA, these arcs being
just equal, and similarly disposetl with respect to the
disturbing forces ; and similarly, that the acceleration
through the arc BC would be exactly compensated by
the retardation along CD. Consequently, on the ave-
rage of each revolution of P, a compensation would take
place ; the period would remain unaltered, and all the
errors in longitude would destroy each other.

(545.) This exact compensation, however, depends
evidently on the exact symmetry of disposal of the parts
of the orbits on either side of the line CSM. If that
symmetry be broken, it will no longer take place, and in-
equalities in P's motion will be produced, which extend
beyond the limit of a single revolution, and must await
their compensation, if it ever take place at all, in a re-
versal of the relations of configuration which produced
them. Suppose, for example, that the orbit of P being
circular, that of M were elliptic, and that, at the moment
when P set out from A, M were at its greatest distance
from P ; suppose, also, that M were so distant as to
make only a small part of its whole revolution during a
revolution of P. Then it is clear that, during the whole
revolution of P, M's disturbing force would be on the
increase by the approach of M, and that, in consequence,
the disturbance arising in each succeeding quadrant of
its motion, would over-compensate that produced in the
foregoing ; so that, when P had come round again to its
conjunction with M, there would be found on the whole
to have taken place an over-compensation in favour of
an acceleration in the orbitual motion. This kind of ac-
tion would go on so long as M continued to approacli S ;

324 A TUEATISt; ON AS-fRONOMY. [cHAP. XI.

but when, in the progress of its elliptic motion, it began
again to recede, the reverse effect would take place, and a
retardation of P's orbitual motion would happen ; and so
on alternately, until at length, in the average of a great
many revolutions of M, in which the place of P in its
ellipse at the moment of conjunction should have been
situated in every variety of distance, and of approach
and recess, a compensation of a higher and remoter order,
among all those successive over and under-compensa-
tions, would have taken place, and a mean or average
angular motion would emerge, the same as if no disturb-
ance had taken place.

(546.) The case is only a little more complicated, but
the reasoning very nearly similar, when the orbit of the
disturbed body is supposed elliptic. In an elliptic orbit,
the angular velocity is not uniform. The disturbed body
then remains in some parts of its revolution longer, in
others for a shorter time, under the inlluence of the ac-
celerating and retarding tangential forces, tlian is neces-
sary for an exact compensation ; independent, then, of
any approach or recess of M, there would, on this account
alone, take place an over or under-compensation, and a
surviving, unextinguished perturbation at the end of a
synodic period ; and, if the conjunctions ahvays took
2)la.ce on the same point of P's ellipse, this cause would
constantly act one way, and an inequality would arise,
having no compensation, and which would at length, and
permanently, change the mean angular motion of P.
But this can never be the case in the planetary system.
The mean motions (i. e. the mean angular velocities) of
the planets in their orbits, are incommensurable to one
another. There are no two planets, for instance, which
perform their orbits in times exactly double, or triple,
the one of the other, or of which the one performs exact-
ly two revolutions while the other performs exactly three,
or five, and so on. If there were, the case in point would
arise. Suppose, for example, that the mean motions of
the disturbed and disturbing planet were exactly in the
proportion of two to five ; then would a cycle, consisting
of live of the shorter periods, or two of the longer, bring
them back exacdy to the same configuration. It would
cause their conjunction, for instance, to happen once in

CHAP. XI.] THEOUY OF JUPITER AND SATURN. 325

every such cycle, in the same precise points of tlieir orbits,
ichile in the, intermediate periods ot" the cycle the other
configurations kept shifting I'ound. TIius, then, Avould
arise the very case we have been contemplating, and a
permanent derangement would happen.

(547.) Now, although it is true that the mean motions
of no two planets are exactly commensurate, yet cases
are not wanting in which there exists an approach to this
adjustment. And, in particular, in the case of Jupiter
and Saturn — that cycle we have taken for our example
in the above reasoning, viz. a cycle composed of five pe-
riods of Jupiter and two of Saturn — although it does not
exactly bring about the same configuration, does so pretty
nearly. Five periods of Jupiter are 21063 days, and two
periods of Saturn 21518 days. The difierence is only
145 days, in which Jupiter describes, on an average, 12°,
and Saturn about 5°, so that after the lapse of the former
interval they will only be 5° from a conjunction in the
same parts of their orbits as before. If we calculate the
time which will exactly bring about, on the average,
three conjunctions of the two planets, we shall find it to
be 21760 days, their synodical period being 7253*4 days.
In this interval Saturn will have descriljed 8° 6' in excess
of two sidereal revolutions, and Jupiter the same angle
in excess of five. Every third conjunction, then, will
take place 8° 6' in advance of the preceding, which is
near enough to establish, not, it is true, an identity with,
but still a great approach to tlie case in question. The
excess of action, for several such triple conjunctions (7
or 8) in succession, will lie tlie same way, and at each
of them the motion of P will be similarly influenced, so
as to accumulate the effect upon its longitude ; thus giv-
ing rise to an irregularity of considerable magnitude and
very long period, which is well known to astronomers
by the name of the great inequality of Jupiter and Saturn.

(548.) The arc 8° 6' is contained 44| times in the
whole circumference of 360° ; and accordingly, if we
trace round this particular conjunction, we shall find it
will return to the same point of the orbit in so many
times 21760 days, or in 2648 years. But the conjunc-
tion we are now considering, is only one out of three
The other two will happen at points of the orbit abn«t

2 E

826 A TREATISE ON ASTRONOMY. [cHAP. XI.

123° and 246° distant, and these points also will advance
by the same arc of 8" 6' in 21760 days. Consequently,
the period of 2648 years will bring them all round, and
in that interval each of them will pass through that point
of the two orbits from which we commenced; hence a
conjunction (one or other of the three) will happen at
that point once in one third of this period, or in 883
years ; and this is, therefore, the cycle in which the
" great inequality" would undergo its full compensation,
did the elements of the orbits continue all that time in-
variable. Their variation, however, is considerable in so
long an interval ; and, owing to this cause, the period
itself is prolonged to about 918 years.

(549.) We have selected this inequality as a proper
instance of the action of a tangential disturbing force,
on account of its magnitude, the length of its period,
and its hiorh historical interest. It had long been re-
marked by astronomers, that on comparing together
modern with ancient observations of Jupiter and Saturn,
the mean motions of these planets did not appear to be
uniform. The period of Saturn, for instance, appeared
to have been lengthening throughout the whole of the
seventeenth century, and that of Jupiter shortening —
that is to say, the one planet was constantly lagging be-
hind, and the other jjettin"- in advance of its calculated
place. On the other hand, in the eighteenth century, a
process precisely the reverse seemed to be going on. It
is true, the whole retardations and accelerations observed
were not very great ; but, as their influence went on
accumvdating, they produced, at length, material differ-
ences between the observed and calculated places of
both these planets, which, as they could not then be ac-
counted for by any theory, excited a high degre of atten-
tion, and were even, at one time, too hastily regarded as
almost subversive of the Newtonian doctrine of gravity.
For a long while this difference baffled every endeavour
to account for it, till at length Laplace pointed out its
cause in the near commensurability of the mean motions,
as above shown, and succeeded in calculating its period
and amount.

(550.) The inequality in question amounts, at its
maximum, to an alternate retardation and acceleration of

CHAP. XI.] TIIEORV OF JUPITER AND SATURN. 327

about 0° 49' in the longitude of Saturn, and a corres-
ponding acceleration or retardation of about 0° 21' in
that of Jupiter. That an acceleration in the one planet
must necessarily be accompanied by a retardation in the
other, and vice versa, is evident, if we consider, that ac-
tion and reaction being equal, and in contrary directions,
whatever momentum Jupiter communicates to Saturn m
the direction PM, the same momentum must Saturn com-
municate to Jupiter in the direction MP. The one, there-
fore, will be dragged forward, whenever the other is
pulled back in its orbit. Geometry demonstrates, that,
on the average of each revolution, the proportion in
which this reaction will affect the longitudes of the two
planets is that of their masses multiplied by the square
roots of the major axes of their orbits, inversely, and this
result of a very intricate and curious calculation is fully
eonfirmed by observation.

(551.) The inequality in question would be much
greater, were it not for the partial compensation which
is operated in it in every triple conjunction of the planets.
Suppose PQR to be Saturn's orbit, and pqr Jupiter's;
and suppose a conjunction to take place at Pp, on the
line SA ; a second at 123° distance, on the line SB ; a
third at 246° distance, on SC ; and the next at 368°, on
SD. This last-mentioned conjunction, taking place
nearly in the situation of the first, will produce nearly a
repetition of the first effect in retarding or accelerating
the planets ; but the other two, being in the most remote
situations possible from the first, will happen under en-
tirely diflerent circumstances as to the position of the
perihelia of the orbits. Now, we have seen that a pre-

328 A TREATISE ON ASTRONOMY. [cHAP. XI.

sentation of the one planet to the other in conjunction,
in a variety of situations, tends to produce compensation ;
and, in fact, the greatest possible amount of compensa-
tion which can be produced by only three configurations
is when they are thus equally distributed round the cen-
tre. Three positions of conjunction compensate more
than two, four than three, and so on. Hence we see
that it is not the whole amount of perturbation, which is
thus accumulated in each triple conjunction, but only
that small part which is left uncompensated by the in-
termediate ones. The reader, who possesses already
some acquaintance with the subject, will not be at a loss
to perceive how this consideration is, in fact, equivalent
to that part of the geometrical investigation of this in-
equality which leads us to seek its expression in terms
of the third order, or involving the cubes and products
of three dimensions of the eccentricities ; and how the
continual accumulation of small quantities, during long
periods, corresponds to what geometers intend when
they speak of small terms receiving great accessions of
magnitude by integration.

(552.) Similar considerations apply to every case of
approximate commensura])ility which can take place
among the mean motions of any two planets. Such, for
instance, is that which obtains between the mean motion
of the earth and Venus — 13 times the period of Venus
being very nearly equal to 8 times that of the earth.
This gives rise to an extremely near coincidence of every
fifth conjunction, in the same j)arts of each orbit (within
^igth part of a circumference), and therefore to a cor-
respondingly extensive accumulation of the resulting un-
compensated perturbation. But, on the other hand, the
part of the pertur])ation thus accumulated is only that
which remains outstanding after passing the equalizing-
ordeal of five conjunctions equally distributed round the
circle ; or, in the language of geometers, is dependent
on powers and products of the eccentricities and inclina-
tions of the fifth onler. It is, therefore, extremely mi-
nute, and the whole resulting inequality, according to
the recent elaliorate calculations of professor Airy, to
whom it owes its detection, amounts to no more than a
few seconds at its maximum, wliile its period is no less

CHAP. XI.] THE moon's VARIATION. 329

than 240 years. This example will serve to show to
what minuteness these inquiries have been carried in the
planetary theory.

(553.) In the theory of the moon, the tangential force
gives rise to many inequalities, the chief of which is that
called the variation, which is the direct and principal
effect of that part of the disturbance arising from the al-
ternate acceleration and retardation of the areas from the
syzigies to the quadratures of the orbit, and vice versa,
combined with the elliptic form of the orbit ; in conse-
quence of wliich, the same area described about the
focus will, in different parts of the ellipse, correspond to
different amounts of angular motion. This inequality,
wliich at its maximum amounts to about 37', was first
distinctly remarked as a periodical correction of the moon's
place by Tycho Brahe, and is remarkable in the history
of the lunar theory, as the first to be explained by New-
ton from his theory of gravitation.

(554.) We come now to consider the effects of that
part of the disturbing force which acts in the direction of
the radius vector, and tends to alter the law of gravity,
and therefore to derange, in a more direct and sensible
manner than the tangential force, the form of the dis-
turbed orbit from that of an ellipse, or, according to the
view we have taken of the subject in art. 536, to produce
a change in its magnitude, eccentricity, and position in
its own plane, or in the place of its perihelion.

(555.) In estimating the disturbing force of M on P,
we have seen that tlie difference only of M's accelerative
attraction on S and P is to be regarded as effective as
such, and that the first resolved portion of iM's attraction,
— that, namely, which acts at P in the direction PS —
not finding in the power which M exerts on P any cor-
responding part, by which its effect may be nullified, is
wholly effective to urge P towards S in addition to its
natural gravity. This force is called the additillous part
of the disturbing force. There is, besides this, another
power, acting also in the direction of the radius SP,
which is that arising from the difference of actions of M
on S and P, estimated first in the direction PL, parallel
to SM, and then resolved into two forces ; one of which
is the tangential force, already considered, in the direction

2e 2

J

330 A TREATISE ON ASTROKOMY. [cHAP. XI.

PK ; the other perpendicular to it, or in the direction PR.
This part of M's action is termed the ablatitioiis force, be-
cause it tends to diminish the gravity of P towards S ; and
it is the excess of tlie one of these resolved portions over
the other, which, in any assigned position of P and M,
constitutes the radial part of the disturbing force, and
respecting whose effects we are now about to reason.
(556.) The estimation of these forces is a matter of no
difficulty when tlie dimensions of the orbits are given,
but they are too complicated in their expressions to find
any place here. It will suffice for our purpose to point
out their general tendency ; and, in the first place, we
shall consider their mean or average efiect. In order to
estimate what, in any one position of P, will be tlie
mean action of M in all the situations it can hold with
respect to P, we have nothing to do but to suppose M
broken up, and distributed in the form of a thin ring
round the circumference of its orbit. If we would take
account of the elliptic motion of M, we might conceive
the thickness of this ring, in its ditferent parts, to be pro-
portional to the time which M occupies in every part of
its orbit, or in the inverse proportion of its angular
motion. But into this nicety we shall not go, but con-
tent ourselves, in the first instance, with supposing M's
orbit circular and its motion uniform. Then it is clear
that the mean disturbing effect on P will be the difference
of attractions of that ring on the two points P and S, of
which the latter occupies its centre, the former is ec-
centric. Now the attraction of a ring on its centre is
manifestly equal in all directions, and therefore, estimated
in any one direction, is zero. On the other hand, on a
point P out of its centre, if unthin the ring, the resulting
attraction will always be outivards, towards the nearest
point of the ring, or directly from the centre.* But if P

• As this is a proposition which the equihbrium of Saturn's ring ren-
ders not merely speculative or illustrative, it will be well to demonstrate
it ; which may be done very simply, and without the aid of any cal-
culus. Conceive a spherical shell, and a point within it: every line
passing through the point, and terminating both ways in the shell, will,
of course, be e(iually inclined to its surface at either end, being a chord
of a spherical surface, and, therefore, symmetrically related to all its
parts. Now, conceive a small double cone, or pyramid, having its apex
at the p<jint, and formed by the conical motion of such a line round the
point. Then will the two portions of the spherical shell, which form the

CHAP. XI. J EFFECTS OF THE RADIAL FORCE. 331

lie without the ring, the resulting force will act always
inwards, urging P towards its centre. Hence it appears
that the mean effect of the radial force will be dilferent
in its direction, according as the orbit of the disturbing
body is exterior or interior to that of the disturbed. In
the former case it will diminish, in the latter will in-
crease, the central gravity.

(557.) Regarding, still, only t!ie mean effect, as pro-
duced in a great number of revolutions of both bodies, it
is evident that an increase of central force must be ac-
companied with a diminution of periodic time, and a
contraction of dimension of the orbit of a body revolving
with a stated velocity, and vice versa. This, then, is the
first and most obvious effect of the radial part of the dis-
turbing force. It alters permanently, and by a certain
mean and invariable amount, the dimensions of all the
orbits and the periodic times of all the bodies composing
the planetary system, from what they would be, did each
planet circulate about the sun uninfluenced by the at-
traction of the rest ; the angular motion of the interior
bodies of the system being thus rendered less, and those
of the exterior greater, than on that supposition. The
latter effect, indeed, might be at once concluded from this
obvious consideration — that all the planets revolving in-
teriorly to any orbit may be considered as adding to the
general aggregate of the attracting matter within, which
is not the less efficient for being distributed over space,
and maintained in a state of circulation.

bases of both the cones, or pyramids, be similar and equally inclined to
their axes. Therefore their areas will be to each other as the squares of
their distances from the common apex. Therefore their attractions on it
will be equal, because the attraction is as the attracting matter directly,
and the square of its distance inversely. Now, these attractions act in
opposite directions, and, therefore, counteract each other. Therefore,
the point is in equilibrium between them ; and as the same is true of
every such pair of areas into which the spherical shell can be broken up,
therefore the point w'ill be in equilibrium, however situated within such a
spherical shell. Now take a ring, and treat it similarly, breaking its
circumference up into pairs of elements, the bases of triangles formed by
lines passing through the attracted point. Here the attracting elements,
being lines, not surfaces, are in the simple ratio of the distances, not the
duplicate, as they should be to maintain the equilibrium. Therefore it
vml not he maintained, but the nearest elements will have the supe-
riority, and the point will, on the whole, be urged towards the nearest
part of the ring. The same is true of every linear ring, and is, therefore,
true of any assemblage of concentric ones forming a flat annulus, like the
ring of Saturn.

332 A TREATISE ON ASTRONOMY. [cHAP. XI.

(558.) This effect, however, is one which we have no
means of measuring, or even of detecting, otherwise than
by calculation. For our knowledge of the periods of
the planets, and the dimensions of their orbits, is drawn
from observations made on tliem in their actual state, and
therefore, under the influence of this constant part of
the perturbative action. Their observed mean motions
are, therefore, affected by tlie whole amount of its in-
fluence ; and we have no means of distinguishing this
from the direct effect of the sun's attraction, Avith which
it is blended. Our knowledge, however, of the masses
of the planets assures us that it is extremely small ; and
this, in fact, is all which it is at all important to us to
know, in the theory of their motions.

(559.) The action of the sun upon the moon, in like
manner, tends, by its mean influence during many suc-
cessive revolutions of both bodies, to dilate permanently
the moon's orbit, and increase her periodic time. But
this general average is not established, either in the case
of the moon or planets, without a series of subordinate
fluctuations due to the elliptic forms of their orbits, which
we have purposely neglected to take account of in the
above reasoning, and whicli obviously tend, in the average
of a great multitude of revolutions, to neutralize each
other. In the lunar theory, however, many of these
subordinate fluctuations are very sensible to observation,
and of great importance to a correct knowledge of her
motions. For example : — The sun's orbit (referred to
the earth as fixed) is elliptic, and requires thirteen luna-
tions for its description, during which the distance of
the sun undergoes an alternate increase and diminution,
each extending over at least six complete lunations.
Now, as the sun approaches the earth, its disturbing
forces of every kind are increased in a high ratio, and
vice versa. Therefore the dilatation it produces on the
lunar orbit, and the diminution of the moon's periodic
time, will be kept in a continual state of fluctuation, in-
creasing as the sun approaches its perigee, and dimi-
nishing as it recedes. And this is consonant to fact — the
observed difference between a lunation in January (when
the sun is nearest the earth) and in July (when it is
farthest) being no less than 35 minutes

CHAP. XX.'] THE MOOX's SECULAR ACCELERATION. 333

(560.) Another very remarkable and important effect
of this cause, in one of its subordinate fluctuations (ex-
tendiufi-, however, over an immense period of time), is
what is called the secular acceleration of the moon's
mean motion. It had been observed by Dr. Halley, on
comparing together the records of the most ancient lu-
nar eclipses of the Chaldean astronomers with those of
modern times, that the period of the moon's revolution
at present is sensibly shorter than at that remote epoch ;
and this result was conlirmed by a further comparison
of both sets of observations with those of the Arabian as-
tronomers of the eighth and ninth centuries. It appear-
ed from these comparisons, that the rate at which the
moon's mean motion increases is about 11 seconds per
century — a quantity small in itself, but becoming consi-
derable by its accumulation during a succession of ages.
This remarkable fact, like the great equation of Jupiter
and Saturn, had been long the subject of toilsome inves-
tigation to geometers. Indeed, so diflicult did it appear
to render any exact account of, that while some were on
the point of again declaring the theory of gravity inade-
quate to its explanation, others were for rejecting altoge-
ther the evidence on which it rested, although quite as
satisfactory as that on which most historical events are
credited. It was in this dilemma that Laplace once more
stepped in to rescue physical astronomy from its re-
proach, by pointing out the real cause of the phenome-
non in question, which, when so explained, is one of the
most curious and instructive in the whole range of our
subject — one which leads our speculations further into
the past and future, and points to longer vistas in the dim
perspective of changes which our system has undergone
and is yet to undergo, than any other which observation
assisted by theory has developed.

(561.) If the solar ellipse were invariable, the alter-
nate dilatation and contraction of the moon's orbit, ex-
plained in art. 559, would in the course of a great many
revolutions of the sun, at length effect an exact com-
pensation in the distance and periodic time of the moon,
by bringing every possible step in the sun's change of
distance to correspond to every possible elongation of
the moon from the sun in her orbit. But this is not, in

334 A TRKATISE ON ASTRONOMY. [cHAP. XI.

fact, the case. The solar eclipse is kept (as we have al-
ready hinted in art. 536, and as we shall very soon ex-
plain more fully) in a continual but excessively slow
state of change, by the action of the planets on the earth.
Its "axis, it is true, remains unaltered, but its eccentricity
is, and has been since the earliest ages, diminishing ;
and this diminution will continue (there is little reason
to doubt) till the eccentricity is annihilated altogether,
and the earth's orbit becomes a perfect circle ; after
which it will again open out into an ellipse, the eccen-
tricity will again increase, attain a certain moderate
amount, and then again decrease. The time required for
these revolutions, though calculable, has not been calcu-
lated, further than to satisfy us that it is not to be reck-
oned by hundreds or by thousands of years. It is a pe-
riod, in short, in which the whole history of astronomy
and of the human race occupies but as it were a point,
during which all its changes are to be regarded as uni-
form. Now, it is by this variation in the eccentricity of
the earth's orbit that the secular acceleration of the moon
is caused. The compensation above spoken of (which,
if the solar ellipse remained unaltered, would be effect-
ed in a few years or a few centuries at furthest in the
mode already stated) will now, we see, be only imper-
fectly effected, owing to this slow shifting of one of the
essential data. The steps of restoration are no longer
identical with, nor equal to, those of change. The same
reasoning, in short, applies, with that by which we ex-
plained the long inequalities produced by the tangential
force. The struggle up hill is not maintained on equal
terms with the downward tendency. The ground is all
the while slowly sliding beneath the feet of the antagonists.
During the whole time that the earth's eccentricity is
diminishing, a preponderance is given to the action over the
reaction ; and it is not till that diminution shall cease, that
the tables will be turned, and the process of ultimate re-
storation will commence. Meanwhile, a minute, outstand-
ing, and uncompensated effect is left at each recurrence,
or near recurrence, of the same configurations of the sun,
the moon, and the solar and lunar perigee. Tliese ac-
cumulate, influence the moon's periodic time and mean
motion, and thus becoming repeated in every lunation,

CHAP. XI. J THE moon's SECULAR ACCELERATION. 335

at length affect her longitude to an extent not to be over-
looked. .

(562.) The phenomenon of which we have now
given an account is another and very striking example of
the propagation of a periodic change from one part of a
system to another. The planets have no direct, appre-
ciable action on the lunar motions as referred to the earth.
Their masses are too small, and their distances too great,
for their difference of action on the moon and earth, ever
to become sensible. Yet their effect on the earth's orbit
is thus, we see, propagated tlirough the sun to that of the
moon ; and what is very remarkable, the transmitted
effect thus indirectly produced on the angle described by
the moon round the earth is more sensible to observa-
tion than that directly produced by them on the angle
described by the eartli round the sun.

(583.) The dilatation and contraction of the lunar and
planetary orbits, then, which arise from the action of the
radial force, and which tend to affect their mean mo-
tions, are distinguishable into two kinds ; — the one per-
manent, depending on the distribution of the attracting
matter in the system, and on the order which each pla-
net holds in it ; the other periodic, and which operates
in length of time its own compensation. Geometers
have demonstrated (it is to Lagrange that we owe this
most important discovery) that, besides these, there ex-
ists no third class of effects, whether arising from the
radial or tangential disturbing forces, or from their com-
bination, such as can go on for ever increasing in one di-
rection without self-compensation ; and, in particular,
that the major axes of the planetary ellipses are not lia-
ble even to those slow secular changes by which the in-
clinations, nodes, and all the other elements of the sys-
tem, are affected, and which, it is true, are periodic, but
in a different sense from those long inequalities which
depend on the mutual configurations of the planets inter
se. Now, the periodic time of a planet in its orbit about
the sun depends only on the masses of the sun and pla-
net, and on the major axis of the orbit it describes, with-
out regard to its degree of eccentricity, or to any other
element. The mean sidereal periods of the planets,
therefore, such as result from an average of a sufficient

336 A TREATISE ON ASTRONOMY. [ciIAP. XI.

number of revolutions to allow of the compensation of
the last-mentioned inequalities, are unalterable by lapse
of time. The length of the sidereal year, for example,
if concluded at this present time from observations em-
bracing a thousand revolutions of the earth round the
sun (such, in short, as we now possess it), is the same
with that which (if we can stretch our imagination so
far) must result from a similar comparison of observa-
tions made a n;iillion of years hence.

(564,) This theorem is justly regarded as the most
important, as a single result, of any which have hitherto
rewarded the researches of mathematicians. We shall,
therefore, endeavour to make clear to our readers, at
least the principle on which its demonstration rests ; and
although the complete application of that principle can-
not be satisfactorily made without entering into details
of calculation incompatible with our objects, we shall
have no difficulty in leading them up to that point where
those details must be entered on, and in giving such an
insio-ht into their general nature as will render it evident
what must be their results when gone through.

(565.) It is a property of elliptic motion performed
under the influence of gravity, and in conformity with
Kepler's laws, that if the velocity with which a planet
moves at any point of its orbit be given, and also the
distance of that point from the sun, the major axis of the
orbit is thereby also given. It is no matter in what
direction the planet may be moving at that moment. This
will influence the eccentricity and the position of its
ellipse, but not its length. This property of elliptic
motion has been demonstrated by Newton, and is one of
the most obvious and elementary conclusions from his
theory. Let us now consider a planet describing an in-
definitely small arc of its orbit about the sun, under the
joint influence of its attraction, and the disturbing power
of another planet. This arc will have some certain cur-
vature and direction, and, therefore, may be considered
as an arc of a certain ellipse described about the sun as
a focus, for this plain reason — that whatever be the
curvature and direction of the arc in question, an ellipse
may always be assigned, whose focus shall be in the sun,
and which shall coincide with it throughout the whole

CHAP. XI.] PERMANENCE OF THE MAJOR AXES. 337

interval (supposed indefinitely small) between its extreme
points. This is a matter of pure geometry. It does not
follow, however, that the ellipse thus instantaneously
determined will have the same elements as that similarly
determined from the arc described in either the previous
or the subsequent instant. If the disturbing force did not
exist, this would be the case ; but, by its action, a vari-
ation of the elements from instant to instant is produced,
and the ellipse so determined is in a continual state of
change. Now, when the planet has reached the end of
the small arc under consideration, the question whether
it will in the next instant describe an arc of an ellipse
having the same or a varied axis will depend, not on the
new direction impressed upon it by the acting forces —
for the axis, as we have seen, is independent of that
direction — not on its change of distance from the sun,
while describing the former arc — for the elements of
that arc are accommodated to it, so that one and the same
axis must belong to its beginning and its end. The
question, in short, whether in the next arc it shall take
up a new major axis, or go on with the old one, will de-
pend solely on this — whether the velocity has undergone,
by the action of the disturbing force, a change incom-
patible with the continuance of the same axis. We say
by the action of tlie disturbing force, because the central
force residing in the focus can impress on it no such
change of velocity as to be incompatible with the per-
manence of any ellipse in which it may at any instant be
freely moving about that focus.

(566.) Thus we see that the momentary variation of
the major axis depends on nothing but tlie momentary
deviation from the law of elliptic velocity produced by
the disturbing force, without the least regard to the
direction in which that extraneous velocity is impressed,
or the distance from the sun at which the planet may be
situated in consequence of the variation of the other
elements of its orbit. And as this is the case at every
instant of its motion, it will follow that, after the lapse
of any time, however great, the amount of change which
the axis may have undergone will be determined by the
total deviation from the original elliptic velocity produced
by the disturbing force ; without any regard to alterations

2F

338 A TREATISE ON ASTRONOMY. [^^HAP. XI.

which the action of that force may have produced in the
other elements, except in so far as the velocity may be
thereby modified. This is the point at which the exact
estimation of the effect must be intrusted to the calcu-
lations of the geometer. We shall be at no loss, how-
ever, to perceive that these calculations can only ter-
minate in demonstrating the periodic nature and ultimate
compensation of all the variations of the axis which can
thus arise, when we consider that the circulation of two
planets about the sun, in the same direction and in in-
commensurable periods, cannot fail to ensure their pre-
sentation to each other in every state of approach and
recess, and under every variety as to their mutual dis-
tance and the consequent intensity of their mutual action.
Whatever velocity, then, may be generated in one by the
disturbing action of the other, in one situation, will in-
fallibly be destroyed by it in another, by the mere efl'ect
of change of configuration.

(567.) It appears, then, that the variations m the
major axes of the planetary orbits depend entirely on
cycles of configuration, like the great inequality of Ju-
piter and Saturn, or the long inequality of the Earth and
Venus above explained, which, indeed, may be regarded
as due to such periodic variations of their axes. In fact,
the mode in which we have seen those inequalities arise,
from the accumulation of imperfectly compensated actions
of the tangential force, brings them directly under the
above reasoning : since the efficacy of this force falls
almost wholly upon the velocity of the disturbed planet,
whose motion is always nearly coincident with or op-
posite to its direction.

(568.) Let us now consider the effect of perturbation
in altering tlie eccentricity and the situation of the axis
of the disturbed orbit in its own plane. Such a change
of position (as we have observed in art. 318) actually
takes place, although very slowly, in the axis of the
earth's orbit, and much more rapidly in that of the
moon's (art. 360) ; and these movements we are now to
account for.

(569.) The motion of the apsides of the lunar and
planetary orbits may be illustrated by a very pretty me-
chanical experiment, which is otherwise insU'Uctive in

CHAP. XI.3 MOTION OF THE APSIDES. 339

giving an idea of the mode in -which orbitual motion is
carried on under the action of central forces variable ac-
cording to the situation of tlae revolving body. Let a
leaden weight be suspended by a brass or iron wire to a
hook in the under side of a firm beam, so as to allow of
its free motion on all sides of the vertical, and so that
when in a state of rest it shall just clear the floor of the
room, or a table placed ten or twelve feet beneath the
hook. The point of support should be well secured
from wagging to and fro by the oscillation of the weight,
which should be sufficient to keep the wire as tightly
stretched as it will bear, with the certainty of not break-
ing. Now, let a very small motion be communicated to
the weight, not by merely withdrawing it from the ver-
tical and letting it fall, but by giving it a slight impulse
sideways. It will be seen to describe a regular ellipse
about the point of rest as its centre. If the weight be
heavy, and carry attached to it a pencil, whose point lies
exactly in the direction of the string, the ellipse may be
transferred to paper lightly stretched and gently pressed
against it. In these circumstances, the situation of the
major and minor axes of the ellipse will remani for a
long time very nearly the same, tliough the resistance of
the air and the stiffness of the wire will gradually di-
minish its dimensions and eccentricity. But if the im-
pulse communicated to the weight be considerable, so as
to carry it out to a great angle (15° or 20° from the
vertical), this permanence of situation of the ellipse will
no longer subsist. Its axis will be seen to shift its
position at every revolution of the weight, advancing in
the same direction with the weight's motion, by an uni-
form and regular progression, Avhich at length will en-
tirely reverse its situation, bringing the direction of the
longest excursions to coincide with that in which the
shortest were previously made ; and so on, round the
whole circle ; and, in a word, imitating to the eye, very
completely, the motion of the apsides of the moon's orbit.
(570.) Now, if we inquire into the cause of this pro-
gi-ession of the apsides, it will not be difficult of de-
tection. When a weight is suspended by a wire, and
drawn aside from the vertical, it is urged to the lowest
point (or rather in a direction at every instant perpen-

340 A TREATISE ON ASTRONOMY. [cHAP. XI.

dicular to the wire) by a force which varies as the sine
of the deviation of the wire from the perpendicular.
Now, the sines of very small arcs are nearly in the pro-
portion of the arcs themselves ; and the more nearly, as
the arcs are smaller. If, therefore, the deviations from
the vertical are so small that we may neglect the curva-
ture of the spherical surface in which the weight moves,
and regard the curve described as coincident with its pro-
jection on a horizontal plane, it will be then moving
under the same circumstances as if it were a revolving
body attracted to a centre by a force varying directly as
the distance ; and, in this case, the curve described would
be an ellipse, having its centre of attraction not in the
focus, but in the centre,* and the apsides of this ellipse
would remain fixed. But if the excursions of the weight
from the vertical be considerable, the force urging it
towards the centre will deviate in its law from the simple
ratio of the distances ; being as the sine, while the dis-
tances are as the arc. Now the sine, though it continues
to increase as the arc increases, yet does not increase so
fast. So soon as the arc has any sensible extent, the sine
begins to fall somewhat short of the magnitude which an
exact numerical proportionality would require ; and
therefore the force urging the weight towards its centre
or point of rest, at great distances falls, in like proportion,
somewhat short of that which would keep the body in its
precise elliptic orbit. It will no longer, therefore, have,
at those greater distances, the same command over the
weight, in proportion to its speed, which would enable

it to deflect it from its rectilinear tangential course into an
ellipse. The true path Mddch it describes will be less

* Newton, Princip. i. 47.

CHAP. XI.] MOTION OF THE APSIDES. 341

curved in the remoter parts than is consistent with the
elliptic figure, as in the annexed cut ; and, therefore, it
will not so soon have its motion brought to be again at
right angles to the radius. It will require a longer con-
tinued action of the central force to do this ; and before
it is accomplished, more than a quadrant of its revolution
must be passed over in angular motion round the centre.
But this is only stating at length, and in a more circuitous
manner, that fact which is more briefly and summarily
expressed by saying that the apsides of its orbit are pro-
gressive.

(571.) Now, this is what takes place, mutatis mu-
tandis, with the lunar and planetary motions. The ac-
tion of the sun on the moon, for example, as we have
seen, besides the tangential force, whose efl^ects we are
not now considering, produces a force in the direction
of the radius vector, whose law is not that of the earth's
direct gravity. When compounded, therefore, with the
earth's attraction, it will deflect the moon into an orbit
deviating from the elliptic figure, being either too much
curved, or too little, in its recess from the perigee, to
bring it to an apogee at exactly 180° from the perigee ;
— too much, if the compound force thus produced de-
crease at a slower rate than the inverse square of the
distance («, e. be too strong in the remoter distances) ;
too little, if the joint force decrease faster than gravity,
or more rapidly than the inverse square, and be therefore
too weak at the greater distance. In the former case,
the curvature, being excessive, will bring the moon to
its apogee sooner than would be the case in an elliptic
orbit ; in the latter, the curvature is insufficient, and will

Fig. 1.

II2;. 2.

^^ "^-v ^^\

/^^^'^^^^''^^^

V ' /

\ /-

therefore bring it later to an apogee. In the former case,
then, the line of apsides will retrograde ; in the latter,
advance. (See^o-. i and^g*. 2.)

(572.) Both these cases obtain in different configura-

f2

342 A TREATISE ON ASTRONOMY. [cHAP. XI.

tions of the sun and moon. In the syzigies, the effect
of the sun's attraction is to weaken the gravity of the
earth by a force, whose law of variation, instead of the
inverse square, follows the direct proportional relation
of the distance ; while, in the quadratures, the reverse
takes place — the whole effect of the radial disturbing
force here conspiring with the earth's gravity, but the
portion added being still, as in the former case, in the
direct ratio of the distance. Therefore the motion of
the moon, in and near the first of these situations, Avill
be performed in an ellipse, whose apsides are in a state
of advance ; and in and near the latter, in a state of re-
cess. But, as we have already seen (art. 556), the ave-
rage effect arising from the mutual counteraction of these
temporary values of the disturbing force gives the pre-
ponderance to the ablatitious or enfeebling power. On
the average, then, of a whole revolution, the lunar apo-
gee will advance.

(573.) The above reasoning renders a satisfactory
enough general account of the advance of the lunar apo-
gee ; but it is not without considerable difiicvdty that it
can be applied to determine numerically the rapidity of
such advance : nor, when so applied, does it account for
the whole amount of the movement in question, as as-
signed by observation — not more, indeed, than about one
half of it ; the remaining part is produced by the tan-
gential force. It is evident, that an increase of velocity
in the moon Avill have the same effect in diminishing the
curvature of its orbit as the decrease of centi-al force,
and vice versa. Now the direct effect of the tangential
force is to cause a fluctuation of the moon's velocity
above and below its elliptic value, and therefore an alter-
nate progress and recess of the apogee. This would
compensate itself in each synodic revolution, were the
apogee invariable. But this is not the case ; the apogee
is kept rapidly advancing by the action of the radial
force, as above explained. An uncompensated portion
of the action of the tangential force, therefore, remains
outstanding (according to the reasoning already so often
employed in this chapter), and this portion is so distri-
buted over the orbit as to conspire with the former cause,
and, in fact, nearly to double its effect. This is what is

CHAP. XI.'] ECCENTRICITIES AND PERIHELIA. 343

meant by geometers, when they say that this part of the
motion of the apogee is due to the square of the disturb-
ing force. The effect of the tangential force in disturb-
ing the apogee would compensate itself, were it not for
the motion which the apogee has already had impressed
upon it by the radial force ; and we have here, therefore,
disturbance reacting on disturbance.

(574.) The curious and complicated effect of pertur-
bation, described in the last article, has given more trou-
ble to geometers than any other part of the lunar theory.
Newton himself had succeded in tracing that part of the
motion of the apogee which is due to the direct action
of the radial force ; but finding the amount only half
what observation assigns, he appears to have abandoned
the subject in despair. Nor, when resumed by his suc-
cessors, did the inquiry, for a very long period, assume
a more promising aspect. On the contrary, Newton's
result appeared to be even minutely verified, and the ela-
borate investigations which were lavished upon the sub-
ject without success began to excite strong doubts whe-
ther this feature of the lunar motions could be explained
at all by the Newtonian law of gravitation. The doubt
was removed, however, almost in the instant of its ori-
gin, by the same geometer, Clairaut, who first gave it
currency, and who gloriously repaired the error of his
momentary hesitation, by demonsti-atlng the exact coin-
cidence between theory and observation, when the effect
of the tangential force is properly taken into the account.
The lunar apogee circulates, as already stated ("art. 360),
in about nine years.

(575.) The same cause which gives rise to the dis-
placement of the line of apsides of the disturbed orbit
produces a corresponding change in its eccentricity.
This is evident on a glance at our figures 1 and 2 of
art. 571. Thus, in fig. 1, since the disturbed body, pro-
ceeding from its lower to its upper apsis, is acted on by
a force greater than would retain it in an elliptic orbit,
and too much curved, its whole course (as far as it is so
affected) will lie ivithin the ellipse, as shown by the
dotted line ; and when it arrives at the upper apsis, its
distance will be less than in the undisturbed ellipse ; that
is to say, the eccentricity of its orbit, as estimated by

344 A TREATISE ON ASTRONOMY. [ciIAP. XI.

the comparative distances of the two apsides from the
focus, will be diminished, or the orbit rendered more
nearly circular. The contrary effect will take place in
the case of fig. 2. There exists, therefore, between the
momentary shifting of the perihelion of the disturbed
orbit, and the momentary variation of its eccentricity,
a relation much of the same kind with that which con-
nects the change of inclination with the motion of the
nodes ; and, in fact, the strict geometrical theories of
the two cases present a close analogy, and lead to final
results of the very same nature. What the variation of
eccentricity is to the motion of the perihelion, the change
of inclination is to the motion of the node. In either
case, the period of the one is also the period of the
other ; and while the perihelia describe considerable an-
gles by an oscillatory motion to and fro, or circulate in
immense periods of time round the entire circle, the ec-
centricities increase and decrease by comparatively small
changes, and are at length restored to their original mag-
nitudes. In the lunar orlnt, as the rapid rotation of the
nodes prevents the change of inclination from accumu-
lating to any material amount, so the still more rapid re-
volution of its apogee effects a speedy compensation in
the fluctuations of its eccentricity, and never suffers
them to go to any material extent ; while the same causes,
by presenting in quick succession the lunar orbit in every
possible situation to all the disturbing forces, whether of
the sun, the planets, or the protuberant matter at the
earth's equator, prevent any secular accumulation of
small changes, by which, in the lapse of ages, its ellip-
ticity might be materially increased or diminished. Ac-
cordingly, observation shows the mean eccentricity of
the moon's orbit to be the same now as in the earliest
ages of astronomy.

(57G.) The movements of the perihelia, and variations
of eccentricity of the planetary orbits, are interlaced
and complicated together in the same manner and nearly
by the same laws as the A'ariations of their nodes and
inclinations. Each acts upon every other, and every
such mutual action generates its own peculiar period of
compensation ; and every such period, in pursuance of
the orinciple of art. 526, is thence propagated throughout

CHAP. XI.3 STABILITY OF THK ECCENTRICITIES, 345

the system. Thus arises cycles upon cycles, of whose
compound duration some notion may be formed, when
we consider what is the length of one such period in the
case of the two principal planets — Jupiter and Saturn.
Neglecting the action of the rest, the etfect of their mu-
tual attraction would be to produce a variation in the ec-
centricity of Saturn's orbit, from 0-08409, its maximinn,
to 0*01.345, its minimum value; while that of Jupiter
would vary between the narrower limits, 0-06036 and
0-02606: the greatest eccentricity of Jupiter correspond-
ing to the least of Saturn, and vice vcrtiu. The period
in which these clianges are gone through, would be 70414
years. After this example, it will be easily conceived
that many millions of years will require to elapse before
a complete fulfilment of the joint cycle which shall re-
store the whole system to its original state as far as the
eccentricities of its orbits are concerned.

(577.) The place of the perihelion of a planet's orbit
is of little consequence to its well-being ; but its eccen-
tricity is most important, as upon this (the axes of
the orbits being permanent) depends the mean tempera-
ture of its surface, and the extreme variations to which
its seasons may be liable. For it may be easily shown
that the mean anmicil amount of light and heat received
by a planet from the sun is, cxteris paribus, as the minor
axis of the ellipse described by it.* Any variation,
therefore, in the eccentricity by changing the minor axi?;,
will alter the 7nean temperature of the surface. How
such a change will also influence the extremes of tempe-
rature appears from art. 315. Now, it may naturally be
inquired whether, in the vast cycle above spoken of, in
which, at some period or other, conspiring changes may
accumulate on the orbit of one planet from several
quarters, it may not happen that the eccentricity of any
one planet — as the earth — may become exorbitantly
great, so as to subvert those -relations which render it
habitable to man, or to give rise to gre^t changes, at least,
in the physical comfort of his state. To this the re-
searches of geometers have enabled us to answer in the
negative. A relation has been demonstrated by Lagrange

* " On the Astronomical Causes which may influence Geological Phe-
nomena."— Geol. Trans. 1832.

346 A TREATISE ON ASTRONOMY. [cHAP. XI.

between the masses, axes of the orbits, and eccentrici-
ties of each planet, similar to what we have already stated
with respect to their inclinations, viz. that if the mass
of each planet be multiplied by the square root of the
axis of its orbit, and the product by the square of its
eccentricity, the sum of all such products throughout
the sy stein is invariable / and as, in point of fact, this
sum is extremely small, so it will always remain. Now,
since the axis of the orbits are liable to no secular changes,
this is equivalent to saying that no one orbit shall in-
crease its eccentricity, unless at the expense of a com-
mon fund, the whole amount of which is, and must for
ever remain, extremely minute.*

(578.) We have hinted, in our last art. but one, at
perturbations produced in the lunar orbit by the protu-
berant matter of the earth's equator. The attraction of
a sphere is the same as if all its matter were condensed
into a point in its centre ; but that is not the case with
a spheroid. The attraction of such a mass is neither
exactly directed to its centre, nor does it exactly follow
the law of the inverse squares of the distances. Hence
will arise a series of perturbations, extremely small in
amount, but still perceptible, in the lunar motions ; by
'which the node and the apogee will be affected. A more
remarkable consequence of this cause, however, is a small
ntltation of the lunar orbit, exactly analogous to that which
the moon causes in the plane of the earth's equator, by its
action on the same elliptic protuberance. And, in gene-
ral, it may be observed, that in the systems of planets
which have satellites, the elliptic figure of the primary
has a tendency to bring the orbits of the satellites to co-
incide with its equator, — a tendency which, though small
in the case of the earth, yet in that of Jupiter, whose el-
lipticity is very considerable, and of Saturn especially,
where the ellipticity of the body is reinforced by the at-
traction of the rings, becomes predominant over every
external and internal cause of disturbance, and produces

* There is nothing in this relation, however, taken per se, to secure
the smaller planets — Mercury, Mars, Juno, Ceres, &c. — from a catas-
trophe, could ihcy accumulate on themselves, or any one of them, the
whole amount of this eccentricity fund. But that can never be : Jupiter
find Saturn will alwa)'s retain the lion's share of it. A similar remark
applies to the inclination fund of art. 515. Theae funds, be it observed,
can never get into debt. Every terra of them is essentially positive.

Masses determined by perturbations. 347

and maintains an almost exact coincidence of the planes
in question. Such, at least, is the case with the nearer
satellites. The more distant are comparatively less af-
fected by this cause, the difference of attractions between
a sphere and spheroid diminishing with great rapidity as
the distance increases. Tlius, while the orbits of all the
six interior satellites of Satui-n lie almost exactly in the
plane of the ring and equator of the planet, that of the
external satellite, whose distance from Saturn is between
sixty and seventy diameters of the planet, is inclined to
that plane considerably. On the other hand, this con-
siderable distance, while it permits the satellite to retain
its actual inclination, prevents (by parity of reasoning)
the ring and equator of the planet from being perceptibly
disturbed by its attraction, or being subjected to any ap-
preciable movements analogous to our nutation and pre-
cession. If such exist, they must be much slower than
those of the earth ; the mass of this satellite (though the
largest of its system) being, as far as can be judged by its
apparent size, a much smaller fraction of that of Saturn
than the moon is of the earth ; while the solar preces-
sion, by reason of the immense distance of the sun, must
be quite inappreciable.

(579.) It is by means of the perturbations of the
planets, as ascertained by observation, and compared
with theory, that we arrive at a knowledge of the masses
of those planets, which, having no satellites, offer no
other hold upon them for this purpose. Every planet
produces an amount of perturbation in the motions of
every other, proportioned to its mass, and to the degree
of advantage or purchase which its situation in the sys-
tem gives it over their movements. The latter is a sub-
ject of exact calculation ; the former is unknown, other-
wise than by observation of its effects. In the determina-
tion, however, of the masses of the planets by this means,
theory lends the greatest assistance to observation, by
pointing out the combinations most favourable for elicit-
ing this knowledge from the confused mass of superposed
inequalities which affect every observed place of a planet;
by pointing out the laws of each inequality in its period-
ical rise and decay ; and by showing how every parti-

348 A TREATISE ON ASTRONOMY. {]CHAP. XI.

cular inequality depends for its magnitude on the mass
producing it. It is thus that the mass of Jupiter itself
(employed by Laplace in his investigations, and inter-
woven with all the planetary tables) has of late been as-
certained, by observations of the derangements produced
by it in the motions of the ultra-zodiacal planets, to have
been insufficiently determined, or rather considerably
mistaken, by relying too much on observations of its sa-
tellites, made long ago by Pound and others, with in-
adequate instrumental means. The same conclusion has
been arrived at, and nearly the same mass obtained, by
means of the perturbations produced by Jupiter on
Encke's comet. The error was one of great importance ;
the mass of Jupiter being by far the most influential ele-
ment in the planetary system, after that of the sun. It
is satisfactory, then, to have ascertained — as by his ob-
servations Professor Airy is understood to have recently
done — the cause of the error ; to have traced it up to its
source, in insufficient micrometric measurements of the
greatest elongations of the satellites ; and to have found
it disappear when measures taken with more care, and
with infinitely superior instruments, are substituted for
those before employed.

(580.) In the same way that the perturbations of the
planets lead us to a knowledge of their masses, as com-
pared with that of the sun, so the perturbations of the
satellites of Jupiter have led, and those of Saturn's at-
tendants will, no doubt, hereafter lead, to a knowledge
of the proportion their masses bear to their respective
primaries. The system of Jupiter's satellites has been
elaborately treated by Laplace ; and it is from his theory,
compared with innumerable observations of their eclipses,
that the masses assigned to them in art. 463 have been
fixed. Few results of theory are more surprising, than
to see these minute atoms weighed in the same balance
which we have applied to the ponderous mass of the
sun, which exceeds the least of them in the enorraous
proportion of 65000000 to 1.

CHAP. XII.3 OF SIDEREAL ASTRONOMY. 349

CHAPTER XII.

OF SIDEREAL ASTRONOMY =

Of the Stars generally — Their Distribution into Classes according to their
apparent Magnitudes — Their Distribution over the Heavens — Of the
Milky Way — Annual Parallax — Real Distances, probable Dimen-
sions, and Nature of the Stars — Variable Stars — Temporary Stars —
Of double Stars — Their Revolution about each Other in elliptic Orbits
— Extension of the Law of Gravity to such Systems — Of coloured
Stare — Proper Motion of the Sini and Stars — Systematic Aberration
and Parallax — Of compound sidereal Systems — Clusters of Stars — Of
Nebute — Nebulous Stars — Annular and planetary Nebulae — Zodiacal
Light.

(581.) Besides the bodies we have described in the
foregoing chapters, the heavens present us with an in-
numerable multitude of other objects, which are called
generally by the name of stars. Though comprehending
individuals differing from each other, not merely in
brightness, but in many other essential points, they all
agree in one attribute — a high degree of permanence as
to apparent relative situation. This has procured them
the title of " fixed stars ;" an expression which is to be
understood in a comparative and not in an absolute sense,
it being certain that many, and probable that all are in a
state of motion, although too slow to be perceptible un-
less by means of very delicate observations, continued
during a long series of years.

(582.) Astronomers are in the habit of distinguishing
the stars into classes, according to their apparent bright-
ness. These are termed magnitudes. The brightest
stars are said to be of the first magnitude ; those which fall
so far short of the first degree of brightness as to make a
marked distinction are classed into the second, and so on
down to the sixth or seventh, which comprise the small-
est stars visible to the naked eye, in the clearest and dark-
est night. Beyond these, however, telescopes continue
the range of visibility, and magnitudes from the 8th down
to the 16th are familiar to those who are in the practice
of using powerful instruments ; nor does there seem the
least reason to assign a limit to this progression ; every
increase in the dimensions and power of instruments,
which successive improvements in optical science have

2G

850 A TREATISE ON ASTRONOMY, [cHAP. XIl.

attained, having brought into view multitutles innumerable
of objects invisible before ; so that, for any thing expe-
rience has hitherto taught us, the number of the stars
may be really infinite, in the only sense in which we can
assign a meaning to the word.

(583.) This classification into magnitudes, however,
it must be observed, is entirely arbitrary. Of a multitude
of bright objects, differing probably, intrinsically, both in
size and in splendour, and arranged at unequal distances
from us, one must of necessity appear the brightest, one
next below it, and so on. An order of succession (rela-
tive, of course, to our local situation among them) must
exist, and it is a matter of absolute indiffereuce, where,
in that infinite progression downwards, from the one
brightest to the invisible, we choose to draw our lines of
demarcation. All this is a matter of pure convention.
Usage, however, has established such a convention, and
though it is impossible to determine exactly, or « priori,
where one magnitude ends and the next begins, and al-
though different observers have differed in their magni-
tudes, yet, on the whole, astronomers have restricted
their first magnitude to about 15 or 20 principal stars;
their second to 50 or 60 next inferior ; their third to
about 200 yet smaller, and so on ; the numbers increas-
ing very rapidly as we descend in the scale of brightness,
the whole number of stars already registered down to the
seventh magnitude, inclusive, amounting to 15000 or
20000.

(584.) As we do not see the actual disc of a star, but
judge only of its brightness by the total impression made
upon the eye, the apparent " magnitude" of any star
will, it is evident, depend, 1st, on the star's distance from
us ; 2d, on the absolute magnitude of its illuminated sur-
face ; 3d, on the intrinsic brightness of thatsurface. Now,
as we know nothing, or next to nothing, of any of these
data, and have every reason for believing that each of
them may differ in different individuals, in the proportion
of many millions to one, it is clear *thal we are not to
expect much satisfaction in any conclusions we may draw
from numerical statements of the number of individuals
arranged in our artificial classes. In fact, astronomers
liave not yet agreed upon any principle by which the

CHAP. XII.] LIGHT OF THE STARS. S51

magnitudes may be pliotometrically arranged, though a
leaning towards a geometrical progression, of which each
term is the half of the preceding, may be discerned.*
Nevertheless, it were much to be wished, that, setting
aside all such arbitrary subdivisions, a numerical estimate
should be formed, grounded on precise photometrical ex-
periments, of the apparent brightness of each star. This
Avould afford a definite character in natural history, and
serve as a term of comparison to ascertain the changes
Avhich may take place in them ; changes whicli we know
to happen in several, and may therefore fairly presume
to be possilile in all. Meanwhile, as a first approxima-
tion, the following proportions of light, concluded from
Sir William Herschel'st experimental comparisons of a
few selected stars, may be borne in mind : —

Light of a star of the average 1st magnitude := 100

2d = 25

3d =12?

4th = 6

5th = 2

6th = 1

By my own experiments, I have found that the light of
Sirius (the brightest of all the fixed stars) is about 324
times that of an average star of the 6th magnitude.^

(585.) If the comparison of the apparent magnitudes
of the stars with their numbers leads to no definite con-
clusion, it is otherwise when we view them in connexion
with their local distribution over the heavens. If indeed
we confine ourselves to the three or four brightest classes,
we shall find them distributed with tolerable impartiality
over the sphere ; but if we take in the whole amount
visible to the naked eye, we shall perceive a great and
rapid increase of number as we approach the borders of
tlie milky way. And when we come to telescopic mag-
nitudes, we find them crowded beyond imagination, along
the extent of that circle, and of the branch which it
sends off from it; so (art. 253) that in fact its whole light
is composed of nothing but stars, Avhose average magni-
tude may be stated at about the tenth or eleventh.

(586.) These phenomena agree witli the supposition

* Struve, Dorpat Catal. of Double Stars, p. xxxv.

t Phil. Tr. 1817. f Trans. Astron. Soc. iii. 183,

353 A TREATISE ON ASTRONOMY. []oHAP. XII,

that the stars of our firmament, instead of being scattered
in all directions indifferently through space, form a stra-
tum, of which the thickness is small, in comparison with
its length and breadth ; and in which the earth occupies
a place somewhere about the middle of its thickness, and
near the point where it sulnlivides into two principal
laminae, inclined at a small angle to each other. For
it is certain that, to an eye so situated, the apparent den-
sity of the stars, supposing them pretty equally scat-
tered through the space they occupy, would be least in
a direction of the visual ray (as SA) perpendicular to
the lamina, and greatest in that of its breadth, as SB, SC,
SD ; increasing rapidly in passing from one to the other
direction, just as we see a slight haze in the atmosphere
thickening into a decided fog bank near the horizon, by
the rapid increase of the mere length of the visual ray.
Accordingly, such is the view of the construction of the
starry firmament taken by Sir William Herschel, whose
powerful telescopes have effected a complete analysis of

this wonderful zone, and demonstrated the fact of its entire-
ly consisting of stars. So crowded are they in some parts
of it, that by counting the stars in a single field of his tele-
scope, he was led to conclude that 50000 had passed under
his review in a zone two degrees in breadth, during a sin-
gle hour's observation. The immense distances at which
the remoter regions must be situated wdll sufficiently ac-
count for the vast predominance of small magnitudes
which are observed in it.

(587.) When we speak of the comparative remote-
ness of certain regions of the starry heavens beyond
others, and of our own situation in them, the question
immediately arises. What is the distance of the nearest
fixed star ? What is the scale on which our visible fir-
mament is constructed ? And what proportion do its di-
mensions bear to those of our own immediate system ?
To this, however, astronomy has hitherto proved unable

CHAP. XII.] DISTANCE OF THE STARS. 353

to supply an answer. All we know on the subject is ne-
gative. We have attained, by delicate observations and
refined coniljinations of theoretical reasoning, to a correct
estimate, first, of the dimensions of the earth ; then,
taking- that as a base, to a knowledge of those of its orbit
about the sun ; and again, by taking our stand, as it were,
on the opposite borders of the circumference of this orbit,
we have extended our measurements to the extreme verge
of our own system, and by the aid of what we know of
the excursions of comets, have felt our way, as it were,
a step or two beyond the orbit of the remotest known
planet. But between that remotest orb and the nearest
star there is a gulf fixed, to whose extent no observa-
tions yet made have enabled us to assign any distinct
approximation, or to name any distance, however im-
mense, which it may not, for any thing we can tell, sur-
pass.

(588.) The diameter of the earth has served us as the
base of a triangle, in the trigonometrical survey of our
system (art. 226), by which to calculate the distance of
the sun : but the extreme minuteness of the sun's paral-
lax (art. 304) renders the calculation from this " ill-
conditioned" triangle (art. 227) so delicate, that nothing
but the fortunate combination of favourable circumstances,
afforded by the transits of Venus (art. 409) could ren-
der its results even tolerably worthy of reliance. But
the earth's diameter is too small a base for direct triangu-
lation to the verge even of our own system (art. 449),
and we are, therefore, obliged to substitute the cmmial pa-
rallax for the diurnal, or, which com.es to the same thing,
to ground our calculation on the relative velocities of the
earth and planets in their orbits (art. 414), when we
would push our triangulation to that extent. It might be
naturally enough expected, that by this enlargement of
our base to the vast diameter of the earth's orbit, the
next step in our survey (art. 227) would be made at a
great advantage ; — that our change of station, from side
to side of it, would produce a perceptible and measurable
amount of annual parallax in the stars, and that by its
means we should come to a knowledge of their distance.
But, after exhausting every refinement of observation, as-
tronomers have been unable to come to any positive and

3 G 2

354 A TREATISE ON ASTRONOMY. £cHAP. XII.

coincident conclusion upon this head ; and it seems,
therefore, demonstrated, that the amount of such paral-
lax, even for the nearest fixed star which has hitherto
been examined with the requisite attention, remains still
mixed up with, and concealed among, the errors inci-
dental to all astronomical determinations. Now, such is
the nicety to which these have been carried, that did the
quantity in question amount to a single second (i. e. did
the radius of the earth's orbit subtend at the nearest fixed
star that minute angle), it could not possibly have escaped
detection and universal recognition.

(589.) Radius is to the sine of 1", in round numbers, as
200000 to 1. In this proportion, then, at least must the
distance of the fixed stars from the sun exceed that of
the sun from the earth. The latter distance, as we have
already seen, exceeds the earth's radius in the proportion
of 24000 to 1 ; and, lastly, to descend to ordinary stand-
ards, the earth's radius is 4000 of our miles. The dis-
tance of the stars, then, cannot be so sma// as 4800000000
radii of the earth, or 19200000000,000 miles ! How much
larger it may be, we know not.

(590.) In such numbers, the imagination is lost. The
only mode we have of conceiving such intervals at all is
by the time which it would require for light to traverse
them. Now light, as we know, travels at the rate of 1 92000
miles per second. It would, therefore, occupy 100000000
seconds, or upwards of three years, in such a journey,
at the very lowest estimate. What, then, are we to
allow for the distance of those innumerable stars of
the smaller magnitude which the telescope discloses to
us ! If 'we admit the lio-ht of a star of each magnitude
to be half that of the magnitude next above it, it will
follow that a star of the first magnitude will require to be
removed to 362 times its distance to appear no larger
than one of the sixteenth. It follows, therefore, that
among the countless multitude of such stars, visible in
telescopes, there must be many whose light has taken at
least a thousand years to reach us ; and that when we
observe their places, and note their changes, we are, in
fact, reading only their history of a thousand years' date,
thus wonderfully recorded. We cannot escape this con-
clusion, but by adopting as an alternative an intrinsic

CHAP. XII.3 INTRINSIC LIGHT OF THE STARS. 355

inferiority of light in all the smaller stars of the milky way.
We shall be better able to estimate the probability of this
alternative, when we have made acquaintance with other
sidereal systems, whose existence the telescope discloses
to us, and whose analogy will satisfy us that the view of
the subject we have taken above is in perfect harmony
with the general tenor of astronomical facts.

(591.) Quitting, however, the region of speculation, and
confining ourselves within certain limits which we are sure
are less than the truth, let us employ the negative know-
ledge we have obtained respecting the distances of the
stars to form some conformable estimate of tlieir real
magnitudes. Of this, telescopes afford us no direct
information. The discs which good telescopes show us
of the stars are not real, but spurious — a mere optical
illusion.* Their light, therefore, must be our only
guide. Now Dr. Wollaston, by direct photometrical
experiments, open, as it would seem, to no objections,!
has ascertained the light of Sirius, as received by us, to
be to that of the sun as 1 to 20000000000. The sun,
therefore, in order that it should appear to us no brighter
than Sirius, would require to be removed to 141400 times
its actual distance. AVe have seen, however, that the dis-
tance of Sirius cannot be so small as 200000 times that of
the sun. Hence it follows, that, upon the lowest possible
computation, the light really thrown out by Sirius cannot
be so little as double that emitted by the sun ; or that
Sirius must, in point of intrinsic splendour, be at least
equal to two suns, and is in all probability vastly greater.:}:

(592.) Now, for what purpose are we to suppose such
magnificent bodies scattered through the abyss of space ?
Surely not to illuminate our nights, which an additional
moon of the thousandth part of the size of our own
would do much better, nor to sparkle as a pageant void
of meaning and reality, and bewilder us among vain
conjectures. Useful, it is true, they are to man as points-
of exact and permanent reference ; but he must have
studied astronomy to little purpose, who can suppose

* See Cab. Cyc. Optics. t Phil. Trans. 1829, p. 21.

X Dr. Wollaston, assuming, as we think he is perfectly justified in do-
ing, a much lower limit ofpoasible parallax in Sirius tlian we have adopts
ed in the text, has concluded the intrinsic light of Sirius to be nearly
that of fourteen suns.

356 A TREATISE ON ASTRONOMY. [cHAP. XII.

man to he the only object of his Creator's care, or who
does not see in the vast and wonderful apparatus around us
provision for other races of animated beings. The planets,
as we have seen, derive their light from the sun ; but that
cannot be ihe case with the stars. These, doubtless, then,
are themselves suns, and may, perhaps, each in its sphere,
be the presiding centre round which other planets, or bo-
dies of which we can form no conception from any ana-
logy offered by our own system, may be circulating.

(59.3.) Analogies, however, more than conjectural, are.
Hot wanting to indicate a correspondence between the
dynamical laws which prevail in the remote regions of
the stars and those which govern the motions of our own
system. Wherever we can trace the law of periodicity —
the regular recurrence of the same phenomena in the
same times — we are strongly impressed with the idea of
rotatory or orbitual motion. Among the stars are se-
veral which, though no way distinguisliable from others
by any apparent change of place, nor by any difference
of appearance in telescopes, yet inidergo a regular period-
ical increase and diminution of lustre, involving, in one
or two cases, a complete extinction and revival. These
are called periodicMl stars. One of the most remarkable
is the star Omicron, in the constellation Cctus, first no-
ticed l)y Fabricius in 159G. It appears about twelve times
in eleven years — or, more exactly, in a period of 334
days ; remains at its greatest brightness about a fort-
night, being then, on some occasions, equal to a large
Star of the second magnitude ; decreases during about
three months, till it becomes completely invisible, in
which state it remains during about five months, when
it again becomes visible, and continues increasing during
the remaining three months of its period. Such is the
general course of its phases. It does not always, how-
ever, return to the same degree of brightness, nor in-
crease and diminish by the same gradations. Hevelius,
indeedj relates (Lalande, art. 79.4) that during the four
years between October, 1072, and December, 1676, it
did not appear at all.

(594.) Another very remarkable periodical star is that
called Algol, or ,3 Persei. It is usually visible as a star
of the second magnitude, and such it continues for the

CHAP. XII.J

PERIODICAL STARS.

357

space of 2'' 14'', when it suddenly begins to diminish in
splendour, and in about 3k hours is reduced to the fourth
magnitude. It then begins again to increase, and in 31
hours more is restored to its usual brightness, going
through all its changes in 2^ 20'' 48™, or thereabouts.
This remarkable law of variation certainly appears
strongly to suggest the revolution round it of some opake
body, which, when interposed between us and Algol,
cuts off a large portion of its light ; and this is accord-
ingly the view taken of the matter by Coodricke, to
whom we owe the discovery of this remarkable fact,* in
the year 1782 ; since which time the same phenomena
have continued to be observed, though with much less
diligence tlian their high interest would appear to merit.
Taken any how, it is an indication of a high degree of
activity, in regions Avhere, but for such evidences, we
might conclude all lifeless. Our own sun requires nine
times this period to perform a revolution on its own axis.
-On the other hand, the periodic time of an opake re-
volving body, sufficiently large, which should produce a
similar temporary obscuration of the sun, seen from a
fixed star, would be less than fourteen hours.

(595.) The following list exhibits specimens of pe-
riodical stars of every variety of period, so far as they can
be considered to be at present ascertained : —

Star's Name.

(3 Persei

y Cephei
/3 LyrEE
<r Aiitinoi
X Herculis
* Serpentis
RA. 1511 4im
PD. 740 15'
0 Ceti
% Cygni
367 B. t Hydrae
34 Fl. Cvffni
420 M. Leonis
X Sagittarii
■i/ Leonis

Period.

2 20 43

5

6

7

60

180

334

37
0

15
0

396 21 0

494

18 years
Many years
Ditto
Ditto

Vari

ation of

Ma

^nitude.

2

to

4

3.4

5

3

4.5

3.4

4.5

3

—

4

7?

—

0

2

0

6

—

n

4

10

6

—

0

7

—

0

3

—

6

6

—

0

6 -

Discoverers.

( Goodricke, 1782.

I Palitzch, 1783.
Goodricke, 1784.
Goodricke, 1784.
Pigott, 1784.
Herschel, 1796.

Harding, 1826.

Fabricius, 1596.
Kirch, 1687.
Maraldi, 1704.
Janson, 1600.
Koch, 1782.
Halley, 1676.
Montanari, 1667.

* See note on page 358.

+ These letters B. Fl. and M. refer to the Cataloguee of Bode, Flam-
eleed, and Mayer

358 A TREATISE ON ASTRONOMY. [cHAP. XII.

The variations of these stars, hov/ever, appear to be
affected, perhaps in duration of period, but certainly in
extent of change, by physical causes at present unknown.
The non-appearance of o Ceti, during four years, has al-
ready been noticed ; and to this instance we may add
that of ;:^^ Cygni, which is stated by Cassini to have been
scarcely visible througliout the years 1G99, 1700, and
1701, at those times when it ought to have been most
conspicuous,

(596.) These irregularities prepare us for other phe-
nomena of stellar variation, which have hitherto been re-
duced to no law of periodicity, and must be looked upon,
in relation to our ignorance and inexperience, as alto-
gether casual ; or, if periodic, of periods too long to have
occurred more than once within the limits of recorded
observation. The phenomena we allude to are those of
temporary stars, which have appeared, from time to time,
in different parts of the heavens, blazing forth with ex-
traordinary lustre ; and after remaining a while appa-
rently immoveable, have died away, and left no trace.
Such is the star which, sitddenly appearing in the year
125 B, C, is said to have attracted the attention of Hip-
parchus, and led him to (h'aw up a catalogue of stars,
the earliest on record. Such, too, was the star which
blazed forth, A, D. 389, near * Aquila;, remaining for
three weeks as bright as Venus, and disappearing entire-
ly. In the years 945, 126*, and 1572, brilliant stars
appeared in the region of the heavens between Cepheus
and Cassiopeia ; and, from the imperfect account we have
of the places of the two eurl^'r, as compared with that of
the last, which was w'ell determined, as well as from the
tolerably near coincidence of the intervals of their appear-
ance, we may suspect them to be one and the same star,
with a period of about 300, or, as Goodricke supposes,

* Th,e s^nio. dist'ovcry appears to have been mnje nearly al)oiit the
saiTie lime by PaHtzch, a fanner of ProlitTr, near Dresden — a peasant by
station, an astronomer by nature — wfip, from his familiar acquaintance
with the aspect of the hea\ ens. bad been h'd to notice among so many
thousand stars lliis one as distinguished from the rest by its variation, and
had ascertained its period. The same Palitzch was also the first to re^
discover the predicted comet of Halley in 1759, vvliidi he saw nearly a
month before any of the astronomers, who, armed with their telescopes,
W'ere anxiously watching its return. These auecjotes carry us back ^o
^he era of the Chaldean shepherds.

CHAP. XII.] TEMPORARY STARS. 359

of 150 years. The appearance of the star of 1512 was
so sudden, that Tycho Brahe, a celebrated Danish astro-
nomer, returning one evening (the 11th of November)
from his laboratory to his dwelling-house, was surprised
to find a group of country people gazing at a star, which
he was sure did not exist half an hour before. This
was the star in question. It was then as bright as
Sirius, and continued to increase till it surpassed Jupiter
when brightest, and was visible at mid-day. It began
to diminish in December of the same year, and in March,
1574, had entirely disappeared. So, also, on the 10th
of October, 1604, a star of this kind, and not less bi'il-
liant, burst forth in the constellation of Serpentarius»
which continued visible till October, 1G05.

(597.) Similar phenomena, though of a less splendid
character, have taken place more recently, as in the case
of the star of the third magnitude discovered in 1670, by
Anthelm, in the head of the Swan ; which, after becom-
ing completely invisible, reappeared, and after under-
going one or two singular fluctuations of light, during
two years, at last died away entirely, and has not since
been seen. On a careful re-examination of the heavens,
too, and a comparison of catalogues, many stars are now
found to be missing ; and although there is no doubt that
these losses have often arisen from mistaken entries, yet
in many instances it is equally certain that there is no
mistake in the observation or entry, and that the star has
really been observed, and as really has disappeare-tl from
the heavens.* This is a branch of practical astronomy
which has been too little followed up, and it is precisely
that in which amateurs of the science, provided with
only good eyes, or moderate instruments, might employ
their time to excellent advantage.! It holds out a sure
promise of rich discovery, and is one in which astrono-

* The star 42 Virginis is inserted in llie Catalogue of the Astronomical
Society from Zach's Zodiacal Catalogue. I missed it on the 9th of Maj',
1828, and have since repeatedly had its place in the field of view of my
20 feet reflector, without perceiving it, unless it be one of two equal stars
of the 9th magnitude, very nearly in the place it must have occupied.— -
Author.

t " Ces variation.? des etoiles sont bien dignes de I'attention desobserv-
ateurs curieux . . . Un jour viendra, peut-etre, oii les sciences auront na*
sez d'amateurs pour qu'on puisse suffire a ccs details." — Lalande, art.
834. — Surely that day is now arrived.

360 A TREATISE ON ASTRONOMY. [cHAP. XU.

mers in established observatories are almost of necessity-
precluded from taking a part by the nature of the ob-
servations required. Catalogues of the comparative
brightness of the stars in each constellation have been
constructed by Sir Wm. Herschel, with the express ob-
ject of facilitating these researches, and the reader will
find them, and a full account of his method of compari-
son, in the Phil. Trans. 1796, and subsequent years.

(598.) We come now to a class of phenomena of quite
a diflerent character, and which give us a real and posi-
tive insight into the nature of at least some among the
stars, and enable us unhesitatingly to declare them subject
to the same dynamical laws, and obedient to the same
power of gravitation, which governs our own system.
Many of the stars, when examined with telescopes, are
found to be double, i. e. to consist of two (in some cases
three) individuals placed near together. This might be
attributed to accidental proximity, did it occur only in a
few instances ; but the frequency of this companionship,
the extreme closeness, and, in many cases, the near equal-
ity of the stars so conjoined, would alone lead to a strong
suspicion of a more near and intimate relation than mere
casual juxtaposition. The bright star Castor, for exam-
ple, when much magnified, is found to consist of two
stars of between the third and fourth magnitude, within
5" of each other. Stars of this magnitude, however,
are not so common in the heavens as to render it at all
likely that, if scattered at random, any two would fall so
near. But this is only one out of numerous such in-
stances. Sir Wm. Herschel has enumerated upwards of
500 double stars, in which the individuals are within half
a minute of each other ; and to this list Professor Struve
of Dorpat, prosecuting the inquiry by the aid of instru-
ments more conveniently mounted for the purpose, has
recently added nearly five times that number. Other ob-
servers have still further extended the catalogue, already
so large, without exhausting the fertility of the heavens.
Among these are great numbers in which the interval be-
tween the centres of the individuals is less than a single
second, of which s Arietis, Atlas Pleiadum, -y Coronas, »
Coronae, « and ^ Herculis, and t and x Ophiuchi, may be
cited as instances. They are divided into classes ac-

EFFECT OF PARALLAX ON A DOUBLE STAR.

361

cording to their distances — the closest forming the first
class.

(599.) When these combinations were first noticed,
it was considered that advantage might be taken of them,
to ascertain whether or not the annual motion of the earth
in its orbit might not produce a relative apparent displace
ment of the individuals constituting a double star. Sup-
posing them to lie at a great distance one behind the other,
and to appear only by casual juxtaposition nearly in the
same line, it is evident that any motion of the earth must
subtend different angles at the two stars so juxtaposed,
and must therefore produce different parallactic displace-
ments of them on the surface of the heavens, regarded
as infinitely distant. Every star, in consequence of the
earth's annual motion, should appear to describe in the
heavens a small ellipse (distinct from that which it would
appear to describe m consequence of the aberration of
ligiit, and not to be confounded with it), being a section,
by the concave surface of the heavens, of an oblique
elliptic cone, having its vertex in the star, and the earth^s
orbit for its base ; and this section will be of less dimen-

sions the more distant is the star. If, then, we regard
two stars, apparently situated close beside each other, but
in reality at very diff'erent distances, their parallactic el-
lipses will be similar, but of diff'erent dimensions. Sup-
pose, for instance, S and s to be tlie positions of two
stars of such an apparently or optically double star a8

2 H

302 A TREATISE ON ASTRONOMY. LcHAP. XII

seen from the sun, and let ABCD, ab c d, be their pa-
rallactic ellipses ; then, since they will be at all times
similarly situated in these ellipses, when the one star
is seen at A, the other will be seen at a. When the
the earth has made a quarter of a revolution in its orbit,
their apparent places will be B^ ; when another quarter,
Cc ; and when another, Drf. If, then, we measure care-
fully, with micrometers adapted for the purpose, their
apparent situation with respect to each other, at different
times of the year, we sliould perceive a periodical change,
both in the direction of the line joining them, and in the
distance between their centres. For the lines A« and Cc
cannot be parallel, nor the lines B6 and J)d equal, unless
the ellipses be of equal dimensions, i. e. unless the two
stars have the same parallax, or are equidistant from the
earth.

(600.) Now, micrometers, properly mounted, enable
us to measure very exactly both the distance between two
objects which can be seen together in the same field of a
telescope, and the position of the line joining them with
respect to the horizon, or the meridian, or any other de-
terminate direction in the licavens. The meridian is
chosen as the most convenient ; and the situation of the
line of junction between the two stars of a double star is
referred to its direction, by placing in the focus of the
eye-piece of a telescope, equatorially mounted, two cross
wires making a right angle, and adjusting their position
so that one of the two stars shall just run along it by its
diurnal motion, while tlie telescope remains at rest; noting
their situation ; and then turning the whole system of
wires round in its own plane by a proper mechaniccd
movement, till the otlicr wire Ijecomes exactly parallel to
their line of junction, and reading off" on a divided circle
the angle the wires have moved through. Such an appa-
ratus is called a position micrometer ; and by its aid we
determine the angle of position of a double star, or the
angle which their line of junction makes with the meri-
dian ; which angle is usually reckoned round the whole
circle, from 0 to 360, beginning at the north and proceed-
ing in the direction north, following (or east) south, pre-
ceding (or west).

(601.) The advantages which this mode of operation

CHAP. XII.] SYSTEMATIC PARALLAX. 363

offers for the estimation of parallax are many and great.
Ill the first place, the result to be obtained, being depend-
ent only on the relative apparent displacement of the two
stars, is unaffected by almost every cause which Avould
induce error in the separate determination of the place
of either by right ascension and declination. Refraction,
that greatest of all obstacles to accuracy in astronomical
determinations, acts equally on both stars ; and is there-
fore eliminated from the result. We have no longer any
thing to fear from errors of graduation in circles from
levels or plumb-lines — from uncertainty attending the
uranographical reductions of aberration, precession, &;c.
' — all which bear alike on both objects. In a word, if we
suppose the stars to have no proper motions of their own
by which a real change of relative situation may arise,
no other cause but their difference of parallax can pos-
sibly affect the observation.

(602.) Such were the considerations which first in-
duced Sir William Herschel to collect a list of double
stars, and to subject them all to careful measurements of
their angles of position and mutual distances. He had
hardly entered, however, on these measurements, before
he was diverted from the original object of the inquiry
(which, in fact, promising as it is, still remains open and
untouched, though the only method which seems to of-
fer a chance of success in the research of parallax) by
phenomena of a very unexpected character, which at
once engrossed his whole attention. Instead of finding,
as he expected, that annual fluctuation to and fro of one
star of a double star with respect to the other — that al-
ternate annual increase and decrease of their distance and
aiigle of position, which the parallax of the earth's an-
nual motion would produce — he observed, in many in-
stances, a regular progressive change ; in some cases
bearing chiefly on their distance — in others on their po-
sition, and advancing steadily in one direction, so as
clearly to indicate either a real motion of the stars them-
selves, or a general rectilinear motion of the sun and
whole solar system, producing a parallax of a higher
order than would arise from the earth's orbitual motion,
and which might be called systematic parallax.

(603.) Supposing the two stars in motion independ-

364 A TREATISE ON ASTRONOMY. [cHAP. XIT.

ently of each other, and also the sun, it is clear that for
the interval of a few years, these motions must be re-
garded as rectilinear and uniform. Hence, a very slight
acquaintance with geometry will suffice to show that the
apparent motion of one star of a double star, referred to
the other as a centre, and mapped down, as it were, on a
plane in which that otlier shall be taken for a fixed or
zero point, can be no other than a right line. This, at
least, must be the case if the stars be independent of
each other; but it will be otherwise if they have a phy-
sical connexion, such as, for instance, real proximity and
mutual gravitation would establish. In that case, they
would describe orbits round each other, and round their
common centre of gravity ; and therefore the apparent
path of either, referred to the other as fixed, instead of
being a portion of a straight line, would be bent into a
curve concave towards that other. The observed mo-
tions, however, were so slow, that many years' observa-
tion was required to ascertain this point ; and it was not,
therefore, until the year 1803, twenty-five years from
the commencement of the inquiry, that any tiling like a
positive conclusion could be come to respecting the rec-
tilinear or orbitual character of the oqservcd changes of
position.

(604.) In that, and the subsequent year, it was dis-
tinctly announced by Sir Willinm Herschel, in two
papers, which wi?l be found in tlie Transactions of the
Royal Society for those years, that there exist sidereal
systems, composed of two stars revolving about each
other in regidar orbits, and constituting what may be
termed binary stars, to distinguish them from double
stars generally so called, in which these physically con-
nected stars are confounded, perhaps, with othei-s only
optically double, or casually juxtaposed in the heavens
at different distances from the eye ; whereas the indi-
viduals of a binary star are, of coui-se, equidistant from
tlie eye, or, at least, cannot differ nrore in distance than
the semidiameter of the orbit they describe about each
other, which is quite insignificant compared witli the
immense distance between tlieni and the earth. Between
fifty and sixty instances of changes, to a greater or less
amount, in the angles of position of double stars, are ad-

CHAP. Xn.3 ELLIPTIC ORBITS OF BINARY STARS. 365

duced in the memoirs above mentioned ; many of which
are too decided, and too regularly progressive, to allow
of their nature being misconceived. In particular, among,
the more conspicuous stars, — Castor, y Virginis, ^ Ursae,
70 Ophiuclii, <r and « Coronae, ^ Bootis, » Cassiopeiae,
y Leonis, ^ Herculis, J Cygni, f^ Bootis, s 4 and s 5 Lyrae,
^ Ophiuclii, ^ Draconis, and ^ Aquarii, are enumerated
as among the most remarkable instances of the observed
motion; aitd'to some of them even periodic times of re-
volution are assigned, approximative only, of course, and
rather to be regarded as roug^h sfuesses than as results of
any exact calculation, for wiiich the data were at the time
quite inadequate. For instance, the revolution of Castor
is set down at 334 years, that of y Virginis at 708, and
that of y Lgonis at 1200 years.

(605.) Subsequent observation has fully confirmed
these residts, not only in their general tenor, but for the
most part in individual detail. Of all the stars above
named, there is not one which is not found to be fully
entitled to be regarded as binary ; and, in fact, this list
comprises nearly all the most considerable objects of that
description which have yet been detected, though (as at-
tention has been closely drawn to the subject, and ob-
servations have multiplied) it has, of late, begun to extend
itself rapidly. The number of double stars which are
certainly known to possess this peculiar character is be-
tween thirty and forty at the time we write, and more
are emerging into notice with every fresh mass of obser-
vations which come before the public. They require
excellent telescopes for their observation, being for the
most part so close as to necessitate the use of very high
magnifiers (such as would be considered extremely
powerful microscopes if employed to examine objects
within our reach), to perceive an interval between the
individuals which compose them.

(606.) It may easily be supposed, that phenomena of
this kind would not pass without attempts to connect
them with dynamical theories. From their first disco-
very, they were naturally referred to the agency of some
power, like that of gravitation, connecting the stars thus
demonstrated to be in a state of circulation about each
other ; and the extension of the Newtonian law of gravi-

2h2

366

A TREATISE ON ASTRONOMY. [CHAP. XII.

tation to these remote systems was a step so obvious, and
so well warranted by our experience of its all-sufficient
agency in our own, as to have been expressly or tacitly
made by every one who has given the subject any share
of his attention. We owe, however, the first distinct
system of calculation, by which the elliptic elements of
the orbit of a binary star could be deduced from observa-
tions of its angle of position and distance at different
epochs, to M. Savary, who showed,* that the motions
of one of the most remarkable among them (| Ursae)
were explicable, within the limits allowable for error of
observation, on the supposition of an elliptic orbit de-
scribed in the short period of 58| years. A different
process of computation has conducted Professor Encket
to an elliptic orbit for 70 Ophiuchi, described in a period
of sevent3'-four years ; and tlu; author of these pages has
himself attempted to contribute his mite to these interest-
ing investigations, 'i'he following may be stated as the
chief results which have been hitherto obtained in this
branch of astronomy : —

Names of Stars.

Teriod of
Rcvolulioii.

Major Suini-
axis of
Ellipse.

Eccentricity.

y liBonis

y Virginia -

(il V.y^m

0- CoronsE

Castor

70 Ophiuchi -

I Ur:3SC

't, Cancri
V, Coronce

Years.
1200
()28-!>()00
452-—
2^()fi0l)U
25-2-r)()00

80-3400

5'-t-2()25

55?

4340

083350

0(>I125
0 75820
040070
041(i4

1-2'0(I0
15430
3-tJ7'J
8-08()
43112
3-857

(607.) Of these, perhaps, the most remarkable is
y Virginis, not only on account of the length of its pe-
riod, but by reason also of the great diminution of ap-
parent distance, and rapid increase of angular motion
about each other, of the individuals composing it. It is
a bright star of tlie fourth magnitude, and its component
stars are almost exactly equal. It has been known to
consist of two stars since the beginning of the eighteenth
century, their distance being then between six and seven
seconds ; so that any tolerably good telescope would re-

* Connois. des Temps, 1830.

t Berlin Ephem. 1838

CHAP. XII. J ELLIPTIC ORBITS OF BINARY STARS. 367

solve it. Since that time they have been constantly ap-
proaching, and arc at present hardly more than a single
second asunder ; so that no telescope, that is not of very
superior quality, is competent to show them otherwise
than as a single star somewhat lengthened in one direc-
tion. It fortunately happens, that Bradley, in 1718, no-
ticed, and recorded in the margin of one of his observa-
tion books, the apparent direction of their line of junction,
as being parallel to that of two remarkable stars, a and S
of the same constellation, as seen by the naked eye ; and
this note, which has been recently rescued from oblivion
by the diligence of Professor Rigaud, has proved of sig-
nal service in the investigation of their orbit. They are
entered also as distinct stars in Mayer's catalogue ; and
this affords also another means of recovering their rela-
tive situation at the date of his observations, which were
made about the year 1756. Without particularising
individual measurements, which will be found in their
proper repositories,* it will suffice to remark, that their
whole series (which since the beginning of the present
century has been very numerous and carefully made, and
which embraces an angidar motion of 100°, and a dimi-
nution of distance to one sixth of its former amount) is
represented with a degree of exactness fully equal to
that of observation itself by an ellipse of the dimensions
and period stated in the foregoing little table, and of
which the further requisite particulars are as follows : —

Perihelion passage. August 18, 1834*

Inclination of orbit to the visual ray ..... 22° 58
Angle of position of the perihelion projected on the heavens 36° 24'
Angle of jiosition of the line of nodes, or intersection of the ) nno 23"
plane of the orbit with the surface of the heavens J

(608.) If the great length of the periods of some of
these bodies be remarkable, the shortness of those of
^Z ers is hardly less so. « Corona) has already made a
complete revolution since its first discovery by Sir Wil-
liam Herschel, and is far advanced in its second period ;
and I Ursae, ^ Cancri, and 70 Ophiuchi, have all accom-
plished by far the greater parts of their respective ellipses
since the same epoch. If any doubt, therefore, could re-
main as to the reality of their orbitual motions, or any

* See them collected in Mem- R. Ast Soc. vol. v. p. 35.

368 A TREATISE ON ASTRONOMY. [cHAP. XII.

idea of explaining thera by mere parallactic changes, these
facts must suffice for their complete dissipation. We
have the same evidence, indeed, of their rotations about
each other that we have of those of Uranus and Saturn
about the sun ; and the correspondence between their
calculated and observed places in such very elongated
ellipses, must be admitted to carry with it a proof of the
prevalence of the Newtonian law of gravity in their sys-
tems, of the very same nature and cogency as that of the
calculated and observed places of comets round the cen-
tral body of our own.

(609.) But it is not with the revolutions of bodies of
a planetary or cometary nature round a solar centre that
we are now concerned ; it is with that of sun around sun
— each, perhaps, accompanied with its train of planets
and their satellites, closely shrouded from our view by
the splendour of their respective suns, and crowded into a
space bearing hardly a greater proportion to the enor-
mous interval which separates them, than the distances
of the satellites of our planets from their primaries bear
to their distances from the sun itself. A less distinctly
characterized subordination would be incompatible with
the stability of their systems, and with the planetary na-
ture of their orbits. Unless closely nestled under the
protecting Aving of their immediate superior, the sweep
of their other sun in its perihelion passage round their
own might carry them oft', or whirl them into orbits ut-
terly incompatible with the conditions necessary for the
existence of their inhabitants. It must be confessed, that
we have here a strangely wide and novel field for specu-
lative excursions, and one which it is not easy to avoid
luxuriating in.

(610.) Many of the double stars exhibit the curious
and beautiful phenomenon of contrasted or complemen-
tary colours.* In such instances, the larger star is usu-
ally of a ruddy or orange hue, while the smaller one ap-
pears blue or green, probably in virtue of that general
law of optics, which provides that when the retina is

* " other suns, perhaps,

With their attendant moons thou wilt descry,
Communicating male and female light,
(Which two great sexes animate tlie world,)
Stored in each orb, perhaps, with some that live.*

Paradise Lost, viii. 148.

CHAP. XII. J COLOIfRED STARS. 369

under the influence of excitement by any bright, coloured
light ; feebler lights, which seen alone would produce
no sensation but of whiteness, shall for the time appear
coloured with the tint complementary to that of the
brigliter. Thus, a yellow colour predominating in the
light of the brighter star, that of the less bright one in the
same field of view will appear blue ; while, if the tint of
the brighter star verge to crimson, that of the other will
exhibit a tendency to green — or even appear as a vivid
green, under favourable circumstances. 'J'he former con-
trast is beautifully exhibited by < Cancri — the latter by y
Andromedae ; both fine double stars. If, however, the
coloured star be much the less bright of the two, it will
not materially affect the other. Thus, for instance, »
Cassiopeiae exhibits the beautiful combination of a large
white star, and a small one of a rich ruddy purple. It is
by no means, however, intended to say, that in all such
cases one of the colours is a mere effect of contrast, and
it may be easier suggested in words, than conceived in
imagination, what variety of illumination two suns — a
red and a green, or a yellow and a blue one — must afford
a planet circulating about either ; and what charming
contrasts and " grateful vicissitudes"' — a red and a green
day, for instance, alternating with a white one and with
darkness — might arise from the presence or absence of
one or other, or both, above the horizon. Insulated stars
of a red colour, almost as deep as that of blood, occur in
many parts of the heavens, but no green or blue star (of
any decided hue) has, we believe, ever been noticed un-
associated with a companion brighter than itself.

(611.) Another very interesting subject of inquiry, in
the physical history of the stars, is their proper motion.
^Ji priori, it might be expected that apparent motions of
some kind or other should be detected among so great a
multitude of individuals scattered through space, and with
nothing to keep them fixed. Their mutual attractions
even, however inconceivably enfeebled by distance, and
counteracted by opposing attractions from opposite quar-
ters, must, in the lapse of countless ages, produce some
movements — some change of intei'nal arrangement — I'e-
sulting from the difference of the opposing actions. And
it is a fact, tliat such apparent motions do exist, not only

370 A TREATISE ON APTRONOMY. [CHAP. XII.

among single, but in many of the double stars ; which,
besides revolving round each other, or round their com-
mon centre of gravity, are transferred, without parting
company, by a progressive motion common to both,
towards some determinate region. For example, the
two stars of 61 Cygni, which are nearly equal, have re-
mained constantly at the same, or very nearly the same,
distance, of 15", for at least fifty years past. Mean-
while they have shifted their local situation in the hea-
vens, in this interval of time, through no less than 4' 23",
the annual proper motion of each star being 5"'3 ; by
which quantity (exceeding a third of their interval) this
system is every year carried bodily along in some un-
known path, by a motion which, for many centuries,
must be regarded as uniform and rectilinear. Among
stars not double, and no way differing from the rest in
any other obvious particular, [j. Cassiopeia? is to be re-
marked as having the greatest proper motion of any yet
ascertained, amounting to 3"*74 of annual displacement.
And a great many others liave been observed to be thus
constantly carried away from their places by smaller, but
not less unequivocal motions.

(612.) Motions which require whole centuries to ac-
cumulate before they produce clianges of arrangement,
such as the naked eye can detect, though quite sufficient
to destroy that idea of mathematical fixity which pre-
cludes speculation, are yet too trifling, as far as practical
applications go, to induce a change of language, and lead
us to speak of the stars in common parlance as otherwise
than fixed. Too little is yet known of their amount and
directions, to allow of any attempt at referring them to
definite laws. It may, however, be stated generally, that
their apparent directions are various, and seem to have
no marked common tendency to one point more than to
another of the heavens. It was, indeed, supposed by Sir
William Herschel, that such a common tendency could
be made out ; and that, allowing for individual deviations,
a general recess could be perceived in the principal stars,
from that point occupied by the star ^ Herculis, towards
a point diametrically opposite. This general tendency
was referred by him to a motion of the sun and solar
system in the opposite direction. No one, who reflects

CHAP. XII.] MOTIONS OF THE SUN AND STARS. 371

with due attention on the subject, will be inclined to deny
the high probability, nay certainty, that the sun has a
proper motion in some direction ; and the inevitable con-
sequence of such a motion, unparticipated by the rest,
must be a slow average apparent tendency of all the stars
to the vanishing point of lines parallel to that direction,
and to the region which he is leaving. This is the ne-
cessary effect of perspective ; and it is certain that it must
be detected by such observations, if we knew accurately
the apparent proper motions of all the stars, and if we
were sure that they were independent, i. e. that the
whole firmament, or at least all that part which we see
in our own neighbourhood, were not drifting along
together, by a general set., as it were, in one direction, the
result of unknown processes and slow internal changes
going on in the sidereal stratum to which our system be-
longs, as we see motes sailing in a current of air, and
keeping nearly the same relative situation with respect
to one another. But it seems to be the general opinion
of astronomers, at present, that their science is not yet
matured enough to afford data for any secure conclusions
of this kind one way or other. Meanwhile, a very in-
genious idea has been suggested by the present astron-
omer royal (Mi*. Pond), viz. that a solar motion, if it
exist, and have a velocity at all comparable to that of
light, must necessarily produce a solar aberration ; in
consequence of which we do not see the stars disposed
as they really are, but too much crowded in the region
the sun is leaving, too open in that he is approaching.
(See art. 280.) Now this, so long as the solar velocity
continues the same, must be a constant effect which ob-
servation cannot detect ; but should it vary, in the course
of ages, by a quantity at all commensurate to the velocity
of the earth in its orbit, the fact would be detected by a
general apparent rush of all the stars to the one or other
quarter of the heavens, according as the sun's motion
were accelerated or retarded ; which observation would
not fail to indicate, even if it should amount to no more
than a very few seconds. This consideration, refined
and remote as it is, may serve to give some idea of the
delicacy and intricacy of any inquiry into the matter of
proper motion ; since the last mentioned effect would ne-

372 A TREATISE ON ASTRONOMY. [cHAP. XII.

cessarily be mixed up with the systematic parallax, and
could only be separated from it by considering that the
nearer stars would be affected more than the distant ones
by the one cause, but both near and distant alike by the
other.

(613.) When we cast our eyes over the concave of the
heavens in a clear night, we do not fail to observe that
there are here and there groups of stars which seem to
be compressed together in a more condensed manner than
in tlie neighbouring parts, forming bright patches and
clusters, which attract attention, as if they were there
brought together by some general cause other than casual
distribution. There is a group, called the Pleiades, in
which six or seven stars may be noticed, if the eye be
directed full upon it; and many more if the eye be turned
carelessly aside, while the attention is kept directed*
upon the group. Telescopes show fifty or sixty large
stars thus crowded together in a very moderate space,
comparatively insulated from the rest of the heavens.
The constellation called Coma Berenices is another such
group, more diffused, and consisting of much larger
stars.

(614.) In the constellation Cancer, there is a some-
what similar but less definite, luminous spot, called
Praesepe, or the bee-hive, which a very moderate tele-
scope— an ordinary night-glass, for instance — resolves
entirely into stars. In the sword handle of Perseus, also,
is another such spot, crowded with stars, which requires
rather a better telescope to resolve into individuals sepa-
rated from each other. These are called clusters of stars ;
and, whatever be their nature, it is certain that other laws
of aggregation subsist in these spots, than those which
have determined the scattering of stars over the general
surface of the sky. This conclusion is still more strongly

• It is a very remarkable fact, that the centre of the visual area is by
far less sensible to feeble impressions of light, than the exterior portions
of the retina. Few persons are aware of the extent to which this com-
parative insensibility extends, previous to trial. To appreciate it, let the
reader look alternately full at a star of the fifth magnitude, and beside it ;
or choose two equally bright, and about 3° or 4° apart, and look full at
one of them, the probability is, he will see only the oilier : such, at least,
is my own case. The fact accounts for the multitude of stars with which
we are impressed by a general view of the heavens ; their paucity
when we come to count them. — Author.

CHAP. XII.] CLUSTERS OF STARS. 375

pressed upon us, when we come to bring very powerful
telescopes to bear on these and similar spots. There are
a gi-eat number of objects which have been mistaken for
comets, and, in fact, have very much the appearance of
comets without tails : small round, or oval nebulous
specks, which telescopes of moderate power only show
as such. Messier has given, in the Connois. des Temps
for 1784, a list of the places of 103 objects of this sort ;
which all those who search for comets ought to be fami-
liar with, to avoid being misled by their similarity of
appearance. That they are not, however, comets, their
fixity sufficiently proves ; and when we come to examine
them with instruments of great power — such as reflectors
of eighteen inches, two feet or more in aperture — any
such idea is completely destroyed. They are then, for
the most part, perceived to consist entirely of stars
crowded together so as to occupy almost a definite out-
line, and to run up to a blaze of light in the centre,
where their condensation is usually the greatest. (See
Jig. 1, pi. ii., which represents (somewhat rudely) the
thirteenth nebula of Messier's list (described by him as
ncbuleuse sans etoiles), as seen in the 20 feet reflector at
Slough.)* Many of them, indeed, arc of an exactly
round figure, and convey the complete idea of a globular
space filled full of stars, insulated in the heavens, and con-
stituting in itself a family or society apart from the re?it,
and subject only to its own internal laws. It would be
a vain task to attempt to count the stars in one of these
globular clusters. They are not to be reckoned by hun-
dreds : and on a rough calculation, grounded on the
apparent intervals between them at the borders (where
they are seen not projected on each other), and the angu-
lar diameter of the whole group, it would appear that
many clusters of this description must contain, at least,
ten or twenty thousand stars, compacted and wedged
together in a round space, whose angular diameter does
not exceed eight or ten minutes ; that is to say, in an
area not more than a tenth part of that covered by the
moon.

* This beautiful object was first noticed by Halley in 1714. It is visi-
ble to the naked eye, between the stars /^ and < llerculis. In a niglit-
glaas it appears exactly like a small round comet,

2J

374 A TREATISE ON ASTRONOMY. [CHAP. XII.

(615.) Perhaps it may be thought to savour of the
gigantesque to look upon tlie imlividuals of such a group
as suns like our own, and their mutual distances as equal
to those wliioli sepai'ate our sun from the nearest fixed
star : yet, when we consider that their united lustre af-
fects the eye with a less impression of light than a star
of the fifth or sixth magnitude (for tlie largest of these
clusters is barely visible to the naked eye), the idea we
are thus compelled to form of their distance from us may
render even such an estimate of their dimensions familiar
to our imagination ; at all events, we can hardly look
upon a group thus insulated, thus in seipso totus, teres,
atque rotundus, as not forming a system of a peculiar
and definite character. Their round figure clearly indi-
cates the existence of some general bond of union in the
nature of an attractive force ; and, in many of tliem,
there is an evident acceleration in the rate of condensa-
tion as we approach the centre, Avliieh is not referable to
a merely uniform distribution of equidistant stars through
a globular space, but marks an intrinsic density in their
state of aggregation greater at tlie centre than at the sur-
face of the mass. It is difficult to form any conception
of the dynamical state of such a system. On tlie one
hand, without a rotatory motion and a centrifugal force,
it is liardly possible not to regard them as in a state of
progressive collapse. On the other, granting such a mo-
tion and such a force, we find it no less difficult to recon-
cile the apparent sphericity of their form with a rotation
of the whole system round any single axis, without which
internal collisions would appear to be inevitable.* The
following are the places, for 1830, of a few of the prin-
cipal of these remarkable objects, as specimens of their
class : —

R.

A.

N. P

. D.

R.

A.

N. P. D.

H.

M.

o

/

K.

M.

O '

13

5

70

5.5

17

29

93 8

13

34

60

45

21

oc>

73 34

15

10

87

16

ai

25

91 34

16

36

53

13

(616.) It is to Sir William Herschel that we owe the
most complete analysis of the great variety of tliose ob-
* Soe a note on tliis subject at the end of tiie work, p- 386.

CHAP. Xn.] OF CLUSTERS OF STARS. 375

jects which are generally classed under the common head
of Nebula, but- which have been separated by him into
— 1st, Clusters of stars, in which the stars are cleai'ly
distinguishable ; and these, again, into globular and ir-
regular clusters ; 2d, Resolvable ne])ula3, or such as ex-
cite a suspicion that they consist of stars, and which
any increase of the optical power of the telescope may
be expected to resolve into distinct stars ; 3d, Nebula?
properly so called, in which there is no appearance
whatever of stars ; which, again, have been subdivided
into subordinate classes, accordingf to their brightness
and size; 4th, Planetary nebulaj ; 5th, Stellar nebula?;
and, 6th, Nebulous stars. The great power of his tele-
scopes has disclosed to us the existence of an immense
number of these objects, and shown them to be distri-
buted over the heavens, not by any means uniformly,
but, generally speaking, with a marked preference to a
broad zone crossing the milky way nearly at right
angles, and whose general direction is not very remote
from that of the hour circle of 0'' and 12''. In some
parts of this zone, indeed — especially where it crosses
the. constellations Virgo, Coma Berenices, and the Great
Kear-^they are assembled in great numbers ; being,
however, for the most part telescopic, and beyond the
reach of any but the most powerful instruments.

(617.) Clusters of stars are either globular, such as
we have already described, or of irregular figure. These
latter are, generally speaking, less rich in stars, and es-
pecially less condensed towards the centre. They are
also less definite in point of outline ; so that it is often
not easy to say where they terminate, or whether they
are to be regarded otherwise tlian as merely richer parts
of the heavens than those around them. In some of them
the stars are nearly all of a size, in others extremely dif-
ferent ; and it is no uncommon thing to find a very red
star much brighter than the rest, occupying a conspi-
cuous situation in them. Sir William Herschel regards
these as globular clusters in a less advanced state of con-
densation, conceiving all such groups as approaching, by
their mutual attraction, to the glolnilar figxn'C, and assem-
bling themselves together from all the surrounding re-
gion, under laws of which we have, it is true, no other

376 A TREATISE ON ASTRONOMY. [cHAP. XII.

proof than the observance of a gradation by which their
characters shade into one another, so that it is impossible
to say where one species ends and the other begins.

(618.) Resolvable nebulae can, of course, only be con-
sidered as clusters either too remote, or consisting of
stars intrinsically too faint to affect us by their individual
Kght, unless Avhere two or three happen to be close
enough to make a joint impression, and give the idea of
a point brighter than the rest. They are almost univer-
sally round or oval — their loose appendages, and irregu-
larities of form, being as it were extinguished by the dis-
tance, and only the general figure of the more condensed
parts being discernible. It is under the appearance of
objects of this character that all the greater globular clus-
ters exhibit themselves in telescopes of insufficient opti-
cal power to show them well ; and the conclusion is
obvious, that those which the most powerful can barely
render resolvable, Avould be completely resolved by a
further increase of instrumental force.

(019.) Of nebula?, properly so called, the variety is
again very great. By far the most remarkable are those
represented in Jigs. 2 and 3, plate II., the former of
which represents the nebulas surrounding the quadruple
(or rather sextuple) star 6 in the constellation Orion ; the
latter, that about «, in the southern constellation Robur
Caroli : the one discovered by Huygens, in 1656, and
figured as seen in the twenty feet reflector at Slough ;
the other by Lacaille, from a figure by Mr. Dunlop, Phil.
Trans. 1827. The nebulous character of these objects,
at least of the former, is very diflerent from what might
be supposed to arise from the congregation of an im-
mense collection of small stars. It is formed of little
flocky masses, like wisps of cloud ; and such wisps
seem to adhere to many small stars at its outskirts, and
especially to one considerable star (represented, in the
figure, below the nebula), which it envelopes with a ne-
bulous atmosphere of considerable extent and singular
figure. Several astronomers, on comparing this nebula
with the figures of it handed down to us by its discoverer,
Huygens, have concluded that its form has undergone a
perceptible change. But when it is considered how dif-
ficult it is to represent such an object duly, and how en-

CHAP. XII.'] OF NEniTL^. 377

tirely its appearance will differ, even in the same tele-
scope, according to the clearness of the air, or other tem-
porary causes, Ave shall readily admit that we have no
evidence of change that can be relied on.

(620.) Plate II. ,^7^^. 3, represents a nebula of a quite
different character. The original of this figiire is in the
constellation Andromeda near the star v. It is visible to
the naked eye, and is continually mistaken for a comet,
by those unacquainted with the heavens. Simon Marius,
who noticed it in 1612, describes its appearance as that
of a candle shining through horn, and the resemblance
is not inapt. Its form is a pretty long oval^ increasing
by insensible gradations of brightness, at first very gra-
dually, but at last more rapidly, up to a central point,
which though very much brighter than the rest, is yet
evidently not stellar, but only nebula in a high state of
condensation. It has in it a few small stars ; but they
are obviously casual, and the nebula itself offers not the
slightest appearance to give ground for a suspicion of
its consisting of stars. It is very large, being nearly
half a degree long, and 15 or 20 minutes broad.

(621.) This may be considered as a type, on a large
scale, of a very numerous class of nebulaj, of a round or
oval figure, increasing more or less in density towards
the central point : they differ extremely, however, in
this respect. In some, the condensation is slight and
gradual ; in others great and sudden : so sudden, indeed,
that they pi'csent the appearance of a dull and blotted
star, or of a star with a slight burr round it, in which
case they are called stellar nebulae ; while others, again,
offer the singularly beautiful and striking phenomenon
of a sharp and brilliant star surrounded by a perfectly
circular disc, or..atmosphere, of faint light in some cases,
dying away on all sides by insensible gradations ; in
others, almost suddenly terminated. These are nebulous
stars. A very fine example of such a star is 55 Andro-
meda R. A. 1** 43"", N. P. D. 50° '/'. s Orionis and / of
the same constellation are also nebulous ; but the nebula
is not to be seen without a very powerful telescope. In
the extent of deviation, too, from the spherical form,
which oval nebula affect, a great diversity is observed :
some are only slightly elliptic ; others much extended

2i2

378 A TREATISE ON ASTRONOMY. [cHAP. XII.

in length ; and in some, the extension so great, as to
give the nebula the character of a long, narrow, spindle-
shaped ray, tapering away at both ends to points. One
of the most remarkable specimens of this kind is in
R.A. 12" 28™; F. P. D. 63° 4'.

(622.) Annular nebulee also exist, but are among the
rarest objects in the heavens. The most conspicuous
of this class is to be found exactly half way between the
stars /2 and y Lyrae, and may be seen with a telescope of
moderate power. It is small, and particularly well de-
fined, so as in fact to have much more the appearance
of a flat oval solid ring than of a nebula. The axes of
the ellipse are to each other in the proportion of about
4 to 5, and the opening occupies about half its diameter:
its light is not quite uniform, but has something of a
curdled appearance, particularly at the exterior edge ;
the central opening is not entirely dark, but is filled up
with a faint hazy light, uniformly spread over it, like a
fine gauze stretched over a hoop.

(6*23.) Planetary nebulae are very extraordinary ob-
jects. They have, as their name imports, exactly the
appearance of planets ; round or slightly oval discs, in
some instances quite sliarply terminated, in others a
little hazy at the borders, and of a light exactly equable
or only a very little mottled, which, in some of them, ap-
proaches in vividness to that of actual planets. What-
ever be their nature, they must be of enormous magnitude.
One of them is to be found in the parallel of v Aquarii,
and about 5™ preceding that star. Its apparent diameter
is about 20". Another, in the constellation Andromeda,
presents a visible disc of 12", perfectly defined and
round. Granting these objects to be equally distant
from us with the stars, their real dimensions must be
such as would fill, on the lowest computation, the whole
orbit of Uranus. It is no less evident that, if they be
solid bodies of a solar nature, the intrinsic splendour of
their surfaces must be almost infinitely inferior to that
of the sun's. A circular portion of the sun's disc, sub-
tending an angle of 20", Avoidd give a light equal to
100 full 71100118; while the objects in question are
hardly, if at all, discernible with the naked eye. The
uniformity of their discs, and their want of apparent

CHAP. Xn.3 PLANETARY NEBULjE. 379

central condensation, woiikl certainly augur their light
to be merely superlicial, and in the nature of a hollow
spherical shell : but whether filled with solid or gaseous
matter, or altogether empty, it would be a waste of
time to conjecture,

(624.) Among the nebula) which possess an evident
symmetry of form, and seem clearly entitled to be re-
garded as systems of a definite nature, however myste-
rious their structure and destination, the most remark-
able are the 51st and 27th of Messier's catalogue. The
former consists of a large and bright globular nebula
surrounded by a double ring, at a considerable distance
from the globe or rather a single ring divided through
about two fifths of its circumference into two laminae,
and having one portion, as it were, turned up out of the
plane of the rest. The latter consists of two bright and
highly condensed round or slightly oval nebulas, united by
a short neck of nearly the same density. A faint nebu-
lous atmosphere completes the figure, enveloping them
both, and filling up the outline of a circumscribed ellipse,
whose shorter axis is the axis of symmetry of the sys-
tem about which it may be supposed to revolve, or the
line passing through the centres of both the nebulous
masses. These objects have never been properly de-
scribed, the instruments with which they were originally
discovered having been quite inadequate to showing the
peculiarities above mentioned, which seem to place them
in a class apart from all others. The one offers obvious
analogies eitiier with the structure of Saturn or with
that of our own sidereal firmament and milky way. The
other has little or no resemblance to any other known
object.

(625.) The nebulae furnish, in every point of view,
an inexhaustible field of speculation and conjecture.
That by far the larger share of them consist of stars
there can be little doubt ; and in the interminable range
of system upon system, and firmament upon firmament,
which we thus catch a glimpse of, the imagination is be-
wildered and lost. On the other hand, if it be true, as,
to say the least, it seems extremely probable, that a phos-
phorescent or self-luminous matter also exists, dissemi-
nated through extensive regions of space, in the manner

380 A TREATISE ON ASTRONOMY. [cHAP. XII.

of a cloud or fog — now assuming capricious shapes, like
actual clouds drifted by the wind, and now concentrating
itself like a cometic atmosphere around particular stars ;
what, we naturally ask, is tlie nature and destination of
this nebulous matter ? Is it absorbed by the stars in
whose neighbourhood it is found, to furnish, by its con-
densation, their supply of liglit and heat ? or is it pro-
gressively concentratmg itself by the eftect of its own
gravity into masses, and so laying the foundation of new
sidereal systems or of insulated stars ? It is easier to
propound such questions than to offer any probable reply
to them. Meanwhile, appeal to fact, by the method of
constant and diligent observation, is open to us ; and, as
the double stars have yielded to this style of questioning,
and disclosed a series of relations of the most intelligible
and interesting description, we may reasonably hope
that the assiduous study of the nebula; will, ere long, lead
to some clearer understanding of their intimate nature.

(626.) We shall conclude this chapter by the men-
tion of a phenomenon which seems to indicate the ex-
istence of some slight degree of nebulosity about the sun
itself, and even to place it in the list of nebulous stars.
It is called the zodiacal light, and may be seen any very
clear evening soon after sunset, about the months of
April and May, or at the opposite season before sunrise,
as a cone or lenticulai'-shaped light, extending from the
horizon obliquely upwards, and following, generally,
the course of the ecliptic, or rather that of the sun's
equator. The apparent angular distance of its vertex
from the sun varies, according to circumstances, from
40° to 90°, and the breadth of its base perpendicular to
its axis from 8° to 30°. It is extremely faint and ill de-
fined, at least in this climate, though better seen in tro-
pical regions, but cannot be mistaken for any atmo-
spheric meteor or aurora boreaiis. It is manifestly in the
nature of a thin lenticulany-formed atmosphere, sur-
rounding the sun, and extending at least beyond the
orbit of Mercury and even of Venus, and may be con-
jectured to be no other than the denser part of that me-
dium, which, as we have reason to believe, resists the
motion of comets ; loaded, perhaps, with the actual ma-
terials of the tails of millions of those bodies, of which

CHAP. XIII.] OF THE CALENDAR. 381

they have been stripped in their successive perihelion
passages (art. 487), and which may be slowly subsiding
into the sun.

CHAPTER XIII.

OF THE CALENDAR.

(627.) Time, like distance, may be measured by com-
parison with standards of" any length, and all that is
requisite for ascertaining correctly the length of any in-
terval, is to be able to apply the standard to the interval
throughout its whole extent without overlapping on the
one hand, or leaving unmeasured vacancies on the other;
to determine, without the possilile error of a unit, the
number of integer standards which the interval admits
of being interposed between its beginning and end ; and
to estimate precisely the fraction over and above an
integer, which remains when all the possible integers are
subtracted.

(628.) But though all standard units of time are equally
possible, theoretically speaking, all are not, practically,
equally convenient. The trppical year and the solar day
are natural units, which the wants of man and the busi-
ness of society force upon us, and compel us to adopt
as our greater and lesser standards for the measurement
of time, for all the purposes of civil life ; and that, in
spite of inconveniences which, did any choice exist,
would speedily lead to the abandonment of one or other.
The principal of these are their incommensurability, and
the want of perfect uniformity in on'e at least of them.

(629.) The mean lengths of the sidereal day and year,
when estimated on an average sufficiently large to com-
pensate the iluctuations arising from nutation in the one,
and from inequalities of configuration in the other, are
the two most invariable quantities which nature presents
us Avith ; the former, by reason of the uniform diurnal
rotation of the earth — the latte? on account of the inva-
riability of the axes of the planetary orbits. Hence it
follows that tlie mean solar day is also invariable. It is

382 A TREATISE ON ASTRONOMY. [cHAP. XIII.

Otherwise with the tropical year. The motion of the
equinoctial points varies not only from the retrograda-
tion of the equator on the ecliptic, but also partly from
that of the ecliptic on the orbits of all the other planets.
It is tlierefore variable, and this produces a variation in
the tropical year, which is dependent on the place of the
equinox (arts, 517, 328). TJie tropical year is actually
above 4*2 P shorter than it was in the time of Hippar-
chus. This absence of the most essential requisite for
a standard, viz. invariability, renders it necessary, since
we cannot help employing the tropical year in our reck-
oning of time, to adopt an arbitrary or artificial value for
it, so near the truth, as not to admit of the accumulation
of its error for several centuries producing any practical
mischief, and thus satisfying the ordinary wants of civil
life ; while, for scientific purposes, the tropical year, so
adopted, is considered only as the representative of a
certain number of integer days and a fraction — the day
being, in effect, the only standard employed. The case
is nearly analogous to the reckoning of value by guineas
and shillings, an arlificial relation of the two coins being
fixed by law, near to, but scarcely ever exactly coincident
with, the natural one, determined by the relative market
price of gold and silver, of which either the one or the
other' — whichever is really the most invariable, or the
most in use with otlier nations — may be assumed as the
true tlieoretical standard of value.

(630.) The other inconvenience of the standards in
question is their incommensurability. In our measure
of space, all our subdivisions are into aliquot parts : a
yard is three feet, a mile eight furlongs, &c. But a year
is no exact number of days, nor an integer number with
any exact fraction, as one third or one fourth, over and
above ; but the surplus is an incommensurable fraction,
composed of hours, minutes, seconds, &c., which pro-
duces the same kind of inconvenience in the reckoning
of time that it would do, in that of money, if we had
gold coins of the value of twenty-one shillings, with odd
pence and farthings, and a fraction of a farthing over. For
this, however, there is no remedy but to keep a strict re-
gister of the surplus fractions ; and, when they amount
to a whole da5^, cast them over into the integer account.

CHAP. XIII.J OF THE CALENDAR. " 383

(631.) To do this in the simplest and most convenient
manner is the object of a well-adjusted calendar. In the
Gregorian calendar, which we follow, it is accomplished,
with remarkable simplicity and neatness, by carrying a
little farther than is done above the principle of an as-
sumed or artificial year, and adopting two such years,
both consisting of an exact integer number of days,
viz. one of 305 and the other of 366, and laying down a
simple and easily remenibered rule for the order in which
these years shall succeed each other in the civil reckoning
of time, so that during the lapse of at least some thou-
sands of years the sum of the integer artificial, or Gre-
gorian, years elapsed shall not differ from the same
number of real tropical years by a whole day. By this
contrivance, the equinoxes and solstices will always fall on
days similarly situated, and bearing the same name, in each
Gregorian year ; and the seasons will for ever correspond
to the same months, instead of running the round of the
whole year, as they must do upon any other system of
reckoning, and used, in fact, to do before this was adopted.

(632.) The Gregorian rule is as follows : — The years
are denominatetl from the birth of Christ, according to
one chronological determination of that event. Every
year whose number is not divisible by 4 without re-
mainder, consists of 365 days ; every year which is so
divisible, but is not divisible by 100, of 366 ; every year
divisible by 100, but not by 400, again of 365 ; and every
year divisible by 400, again of 360. For example, the
year 1833, not being divisible by 4, consists of 365
days; 1836 of 366; 1800 and 1900 of 365 each; but
2000 of 366. In order to see how near this rule will
bring us to the truth, let us see what number of days
10000 Gregorian years will contain, beginning with the
year 1. Now, in 10000, the numbers not divisible by 4
will be I of 10000, or 7500 ; those divisible by 100, but
not by 400, will in like manner be | of 100, or 75 ; so
that, in the 10000 years in question, 7575 consists of
366, and the remaining 2425 of 365, producing in all
3652425 days, which would give for an average of each
year, one with another, 365'^'2425. The actual value of
the tropical year (art. 327) reduced into a decimal frac-
tion, is 305-34224, so the error of the Gregorian rule on

384 A TREATISE ON ASTRONOMY. [CHAP, XIII.

10000 of the present tropical years is 2*0, or 2'' M*" 24™ ;
that is to say, less than a day in 3000 years ; which is
more than sufficient for all human purposes, those of the
astronomer excepted, who is in no danger of being led
into error from this cause. Even this error might be
avoided by extending the wording of the Gregorian rule
one step farther than its contrivers probably thought it
worth while to go, and declaring that years divisible by
4000 should consist of 365 days. This would take off
two integer days from the above calculated nundjer, and
2-5 from a larger average ; making the sum of days in
100000 Gregorian years, 36524225, which differs only
by a single day from 100000 real tropical years, such as
they exist at present.

(633.) As any distance along a high road might,
though in a rather inconvenient and roundabout way, be
expressed without introducing eri'or by setting up a series
of milestones, at intervals of unequal lengths, so that
every fourth mile, for instance, should be a yard longer
than the rest, or according to any other fixed rule ; taking
care only to mark the stones, so as to leave room for no
mistake, and to advertise all travellers of the difference
of lengths and their order of succession ; so may any in-
terval of time be expressed correctly by stating in what
Gregorian years it begins and ends, and whereabouts in
each. For this statement, coupled with the declaratory
rule, enables us to say how many integer years are to be
reckoned at 365, and how many at 366 days. The latter
years are called bissextiles, or leap-years, and the sur-
plus days thus thrown into the reckoning are called in-
tercalary or leap-days.

(634.) If the Gregorian rule, as above stated, had al-
ways been adhered to, nothing would be easier than to
reckon the number of days elapsed between the present
time and any historical recorded event. But this is not
the case ; and the history of the calendar, with reference
to chronology, or to the calculation of ancient observa-
tions, may be compared to that of a clock, going regularly
when left to itself, but sometimes forgotten to be wound
up ; and when wound, sometimes set forward, sometimes
backward, and that often to serve particular purposes and
private interests. Such, at least, appears to Lave been

CHAP. XIII.] OF THE CALENDAR. 385

tlie case with the Roman calendar, in which our own
originates, from the time of Numa to that of Julius
Caesar, Avhen the lunar year of 13 months, or 355 days,
was augmented at pleasure, to correspond to the solar,
by which the seasons are determined, by the arbitrary
intercalations of the priests, and the usurpations of the
decemvirs and other magistrates, till the confusion be-
came inextricable. To Julius Caesar, assisted by Sosi-
genes, an eminent Alexandrian astronomer and mathe-
matician, we owe the neat contrivance of the two years
of 365 and 366 days, and the insertion of one bissextile
after three common years. This important change took
place in the 45th year before Christ, which was the first
regular year, commencing on the 1st of January, being
the day of the new moon immediately following the
winter solstice of the year before. We may judge of
the state into which the reckoning of time had fallen,
by the fact, that, to introduce the new system, it was
necessary to enact that the previous year (46 b. c.)
sliould consist of 455 days, a circumstance which ob-
tained it the epithet of " the year of confusion."

(635.) The Julian rule made every fourth year, witli-
out exception, a bissextile. This is, in fact, an over-
correction ; it supposes the length of the tropical year to
be 3654*1, which is too great, and thereby induces an
error of 7 days in 900 years, as will easily appear on
trial. Accordingly, so early as the year 1414, it began
to be perceived that the equinoxes were gradually creep-
ing away from the 21st of March and September, where
they ought to have always fallen had the Julian year
been exact, and happening (as it appeared} too early.
The necessity of a fresh and effectual reform in the calen-
dar was from that time continually urged, and at length
admitted. The change (which took place under the
popedom of Gregory XIII.) consisted in the omission of
ten nominal days after the 4th of October, 1582 (so that
the next day was called the 15th, and not the 5th), and
the promulgation of the rule already explained for future
regulation. The change was adopted immediately in all
catholic countries ; but more slowly in protestant. In
England, " the change of style," as it was called, took
place after the 2d of September, 1752, eleven nominal

2K

386 A TREATISE ON ASTRONOMY. []CHAP. Xlll.

(lays being then struck out ; so that, the last day of Old
Style being the 2d, the first of New Style (the next day)
was called the 14th, instead of the 3d. The same legis-
lative enactment which established the Gregorian yeai
in England in 1753, shortened the preceding year, 1751,
by a full quarter. Previous to that time, the year was
lield to begin with the 25th March, and the year a. d.
1751 did so accordingly ; but that year was not suffered
to run out, but was supplanted on the 1st January by
the year 1752, which it was enacted should com-
mence on that day, as well as every subsequent year.
Russia is now the only country in Europe in which tlie
Old Style is still adhered to, and (another secular year
having elapsed) tlie difference between the European and
Russian dates amounts, at present, to 12 days.

(6.S6.) It is fortunate for astronomy that the confusion
of dates and the irreconcilable contradictions which his-
torical statements too often exhibit, when confronted
with the best knowledge we possess of tlie ancient reck-
onings of time, affect recorded observations but little. An
astronomical observation, of any striking and well marked
phenomenon, carries with it, in most cases, abundant
means of recovering its exact date, when any tolerable ap-
proximation is afforded to it by chronological records ;
and, so far from being abjectly dependent on the ob-
scure and often contradictory dates which the compari-
son of ancient authorities indicates, is often itself the
surest and most convincing evidence on which a chrono-
logical epoch can be brought to rest. Remarkable eclipses,
for instance, now that the lunar theory is thoroughly un-
derstood, can be calculated back for several thousands of
years, without the possibility of mistaking the day of
their occurrence. And whenever any such eclipse is so
interwoven with the account given by an ancient author
of some historical event, as to indicate precisely the
interval of time between the eclipse and the event, and
at the same time completely to identify the eclipse, that
date is recovered and fixed for ever.*

(637.) The days thus parcelled out into years, the

* See the remarkable calculations of Mr. Baily relative to the cele-
brated solar ecliiise which put an end to tlie battle between tlie kings
of Metlia and Lydia, b. c GIO, Sept. '30. Phil. Trans, ci. 220.

CHAP. XIII. J OF THK CALENDAR. 387

next step to a perfect knowledge of time is to secure the
idcntitication of each day, by imposing on it a name uni-
versally known and employed. Since, however, the
days of a whole year are too numerous to admit of load-
ing tlie memory with distinct names for each, all nations
have felt the necessity of breaking them down into par-
cels of a more moderate extent ; giving names to each
of these parcels, and particularizing the days in each by
numbers, or by some especial indication. The lunar
month has been resorted to in many instances ; and some
nations have, in fact, preferred a lunar to a solar chro-
nology altogether, as the Turks and Jews continue to do
to this day, making the year consist of 13 lunar months,
or 355 days.* Our own division into twelve unequal
months is entirely arbitrary, and often productive of con-
fusion, owing to the equivoque between the lunar and
calendar month. The intercalary day naturally attaches
itself to February as the shortest.

* The Metonio cycle, or llie fact, discovered by Meton, a Greek ma-
thematician, that 19 solar years contain just 235 lunations (which in fact
they do to a very great degree of approximation), was duly appreciated
by the Greeks, as ensuring tlie corresiwndence of the solar and lunar
years, atid honours were decreed to its discoverer.

388 A TREATISE ON ASTRONOMY.

NOTE

On Uie Constitution of a Globular Cluster, referred to tn page 374.

If we suppose a globular space filled with equal stars, uniformly dis-
pei-sed through it, and very numerous, each of them attracting every
other with a force inversely as the square of the distance, the resultant
force by which any one of them (those at the surface alone excepted)
will be urged, in virtue of their joint attractions, will be directed towards
the common centre of the sphere, and will be directly as the distance
therefrom. This follows from what Newton has proved of the internal
attraction of a homogeneous sphere. Now, under such a law of force,
each particular star would describe a perfect ellipse about the common
centre of gravity, as its centre, and that, in whatever plane and whatever
direction it might revolve. The condition, therefore, of a rotation of
the cluster, as a mass, about a single axis would be unnecessary. Each
ellipse, whatever might be the proporlion of its axes, or the inclination of
its plane to the others, would be invariable in every particular, and all
would be described in one common period, so that at the end of every
such period or annus magnus of the system, every star of the cluster
(except the superficial ones) would be exactly re-established in its
original position, thence to set out afresh and run the same unvarying
round for an indefinite succession of ages. Supposing their motions,
therefore, to be so adjusted at any one moment as that the orbits
should not intersect each other, and so that the magnitude of each star,
and the sphere of its more intense attraction, should bear but a small pro-
portion to the distance separating the individuals, such a system, it is
obvious, might subsist, and realize, in great measure, that abstract and
ideal harmony, which Newton, in the 89th Proposition of the First Book
of the Frincipia, has shown to characterize a law of force directly as
the distance. See also Quarterly Review, No. 94, p. 540. — Author-

A TREATISE ON ASTRONOMY.

389

Stnoptic Table or the Elements of tue Solab, Sistem.

N. B.— The data for Vesta, Juno, Ceres, and Pallas are for January 1, 1820.
The rest for January 1, 1801.

Planet's

Mean distance

Mean Sidereal

Eccentricity in

from Sun, or

Period in Mean

Parts of the

Semi-axis.

Solar Days.

Semi-axis.

Mercury

0-3870981

87-9692580

0-2055149

Venus

0-7233316

224-7007869

0-0068607

Earth

1-0000000

365-2563612

0-0167836

Mars

1-5236923

686-9796458

0-0933070

Vesta

2-3678700

1325-7431000

0-0891300

Juno

2-6690090

1592-6608000

0-2578480

Ceres

2-7672450

1681-3931000

0-0784390

Pallas

2-7728800

1686-5388000

0-2410480

Jupiter

5-2027760

4332-5848212

0-0481621

Saturn

9-5387861

10759-2198174

0-0561505

Uranus

19-1823900

30686-8208296

0-0406794

Planet's

Inclination to the

Loncitude of

Longitude of

name.

Ecliptic.

ascending Node.

Perihelion.

Mercury

7° 0' 9"-l

45° 57' 30" -9

74° 21' 46" -9

Venus

3 23 28 -5

74 54 12 -91128 43 53 -ll

Earth
Mars

99 30 5 -0
332 23 56 -6

1 51 6 -2

48 0 3-5

Vesta

7 8 9-0

103 13 18 -2 249 33 24 -41

Juno

13 4 9-7

171 7 40 -4

53 33 46 -0

Ceres

10 37 26 -2

80 41 24 -0 147 7 31 -5

Pallas

34 34 55 -0

172 39 26 -8121 7 4 -3

Jupitur

1 18 51 -3

98 26 18 -9

11 8 34 -6

Saturn

2 29 35 -7

111 50 37 -4

89 9 29 -8

Uranus

0 46 28 -4

72 59 35 -3

167 31 16 -1

Planet's
name.

Mean Longitude
at the Epoch.

Mass in Billionths
of the Sun's.

Equatorial Dia-
meter, the Sun's
being«lil-454.

Mercury

166° 0' 48" -6

493628

0-398

Venus

11 33 3 -0

2463836

0-975

Earth

100 39 10 -2

2817409

1-000

Mars

64 22 55 -5

392735

0-517

Vesta

Juno

Ceres

Pallas

Jupiter

278 30 0 -4
200 16 19 -1
123 16 11 -9
108 24 57 -9
112 15 23 -0

953570222

10-860

Saturn

135 20 6 -5

284738000

9-987

Uranus

177 48 23 -0

55809812

4-332

2 k2

390

A TREATISE ON ASTRONOMY.

Synoptic Table of the Elements of the Orbits of
THE Satellites, so far as they are known.

N. B. — The distances are expressed in equatorial radii of the pri-
maries. The epoch is Jan. 1, 1801. The periods, &c. are ex-

pressed in mean solar days.

I. The Moon.

Mean distance from earth
Mean sidereal revolution
Mean synodical ditto
Eccentricity of orbit
Mean revolution of nodes
Mean revolution of apogee
Mean longitude of node at epoch
Mean longitude of perigee at do.
Mean inclination of orbit
Mean longitude of moon at epoch
Mass, that of earth being 1, .
Diameter in miles

29'-982 17500
27'^-3216614]8
29''-530588715
0-054844200
6793d-391080
3232'*-575343

13° 53' 17"

266 10 7

5 8 47

118 17 8

0-0125172

2160

•7
•5
•9
•3

II. Satellites of Jupiter.

Sat.

Mean Distance.

Sidereal
Revolution.

Inclination of

Orbit to that of

Jupiter.

Mass; that
of Jupiter

being
1000000000.

1
2
3

4

6-04853

9-62347

15-35024

26-99835

■ l^ 18^ 28""

3 13 14

7 3 43

16 16 32

3° 5' 30"

Variable

Variable

2 58 48

17328
23235
88497
42659

The eccentricities of the 1st and 2d satellite are insensible, that
of the 3d and 4th small, but variable in consequence of their mutual
perturbations.

A TREATISE ON ASTRONOMY.

391

III. Satellites of Saturn.

1
Sat.

Mean
Distance

Sidereal
Revolution.

Eccentricities and Inclinations.

1
2
3

4
5
6

7

3-351
4-300
5-284
6-819
9-524
22-081
64-359

0-1 22'^ 38™
1 8 53

1 21 18

2 17 45
4 12 25

15 22 41
79 7 55

The orbits of the six interior
satelhtes are nearly circular,
and very nearly in the plane of
the ring. That of the seventh
is considerably inclined to the
rest and approaches nearer to
coincidence with the ecUptic.

IV. Satellites of Uranus.

Sat.

Mean
Distance.

Sidereal Period.

Inclination to Ecliptic.

V.

13-120

5a

21h 25'" 0^

Their orbits are inclined

2

17-022

8

16 56 5

about 78° 58' to the

3?

19-845

10

23 4 0

ecliptic, and their motion

4

22-752

13

11 8 59

is retrograde. The pe-

5?

45-507

38

1 48 0

riods of the 2d and 4th

61

91-008

107

16 40 0

require a trifling correc-
tion. The orbits appear
to be nearly circles.

T^l-

Fui. Z.

Fig. 3.

Vihu'tr Sattp.

riate Z^

'•Jwl

Fi.-. ^

J.ylaoer .

Iwl

ho i.

Vk .'.

INDEX.

Air, 28. Mechanical laws for regu-
lating its dilation and compres-
sion ; rarefraction of, 29. Density
of, 29. Refractive power of, af-
fected by its moisture, 33.

Angle of reflection equal to that of
incidence, 91.

Angles, measurement of, 82.

Anomalistic and tropical years, 196.

Apparent diurnal motion of the hea-
venly bodies explained, 45.

Apsides, their motion illustrated,
338.

Astronomical instruments, 66.
Practical difficulties in the con-
struction of 67. Observations in
general, 68.

Astronomy, 7. General notions
concerning the science, 14.

Atmosphere, 29. Refractive power
of the, 31. General notions of its
amount, and law of variation, 34.
Reflective power of 36.

Attraction, magnetic and electric,
224. Of spheres, 225. Solar at-
traction, 227.

Azimuth and altitude instruments,
99.

B.

Barometrical determination of
heighls, 149.

Biot, M., his aeronautic expedition,
27,

Bode's law of planetary distances,
262.

Bodies, effect of the earth's attrac-
tion on, 124. Motion of, 222.
Rule for determining the velo-
city of 223. Problem of three,
296.

Borda, his invention of the principle
of repetition, 103.

Calendar, 381. Gregorian, 383.
Juhan, 385.

Cause and effect, 221 .

Celestial refraction, 38. Maps, 151.
Construction of, by observationa
on right ascension and declina-
tion, 152. Objects divided into
fixed and erratic, 155. Longitudes
and latitudes, 160.

Centrifugal force, 118.

Chronometers, 78.

Circles, co-ordinate, 96.

Clairaut, 124.

Clepsydras, 78.

Clocks, 78.

Comets, their number, 283. Their
tails, 284. Their constitution, 285.
Their orbits, 287. Their predicted
returns; Encke's, 291. Biela's,
291. Their dimensions, 293.

Copernican explanation of the sun'a
apparent motion, 185.

Dates, astronomical meaiia of fixing,
386.

Day, solar, civil measure of time,
381. Sidereal, 381.

Definitions of various terms employ-
ed in astronomy, 56.

Diurnal or geocentric parallax, 181.

E.

Earth, the, one of the principal ob-
jects of the astronomer's conside-
ration; opinions of the ancients
concerning, 16. Real and appa-
rent . motion of explained, 17.
Form and magnitude of 19. Its
apparent d iameter, 21 . A diagram
elucidating the circular form of,
22. Effect of the curvature of,
24, Diurnal rotation of, 42. Poles

393

394

INDEX.

of, 50. Figure of, lOG. Means--
of determining with accuracy tlie
dimensions of the whole or any
part of, explained, 107. Meridio-
nal section of, 112. Exact dimen-
sions of, 114. Its form that of
erjuilibrium, modilied by centri-
fugal force, 117. Local variation
of gravity on its surface, 120. Ef-
fects of the earth's rotation, 123.
Correction for the sphericity of
144. Tlie point of the earth's
axis, 103. Conical movements of,
164. Mutation of, Ifiij. Parallel-
ism of 186. Proportion of its mass
to that of the sun, 274.

Ecliptic, the, 157. Its position among
the stars, 158. Poles of 159.
Plane of its secular variation, 308.

Elliptic motion, laws of 179.

Equations for precession and nuta-
tion, 167.

Equatorial or parallactic instru
ment, 98.

Equinoxes, precession of the, 162
Uranographical effect of 162.

Eccentricity ol'the planetaiy orbits
its variation, 343.

Explanation of the seasons, 186.

Floating collimator, invented by

Captain Kater, 95.
Force, centrifugal, 223

G.

Gay-Lussac, his aeronautic expedi-
tion, 28.

Galileo discovers Jupiter's satellites,
279.

Geographical latitudes determined,
129.

Geography, outline of so far as it is
to bo considered a part of astro-
nomy, 105.

Gravitation, law of universal, 222.

Gravity, local, variation of 119.
Statical measure of 121. Dyna-
mical measure of 122. Terres-
trial, 222. Diminution of, at the
moon, 224. Solar, 229.

H.

Hadley's sextant, 101.

Halley discovers the secular accele-

ration of the moon's mean motion,

333.
Harding, Professor, 262.
Herschel, Sir William, his view of

tlie physical constitution of the

sun, 198.
Horizon, dip of the, explained, 23.
Hour-glass, 78.

K.

Rater's floating collimator, 95.

Kepler, the first who ascertained the
elliptic form of the earth's orbit,
179. His laws, and their inter-
pretation, 250,

L.

Lalande, his ideas of the spots on
the sun, 199.

Laplace accounts for the secular ac-
celeration of the moon, 333.

Latitude, 59. Length of a degree
of, 109.

Level, description and use of 92.

Light, aberration of 169. Urano-
graphical effect of 172. Its velo-
city proved by eclipses of Jupiter's
satellites, 280.

Longitudes, determination of by
astronomical observation, 131.
Differences found by chronome-
ters, 132. Determined by tele-
graphic signals, 134.

Lunar eclipses, 215.

M.

Maclaurin, 124.

Maps, construction of 141. Projec-
tions chiefly used in, 146. The
orthographic, stereographic, and
Mercator's, 146.

Menstrual equation, 273.

Mercator's projection of the sphere,
147.

Mercury, the most reflective fluid
known, 91.

Meridian, or transit circle, for ascer-
taining the right ascensions and
polar distances of objects, 91.

Microscope, compound, 85.

Milky Way, 157, 351.

Moon, the, its sidereal period; its
apparent diameter, 203. Its paral-
lax, distance, and real diameter

INDEX.

395

204. The I'orm of its orbit, liki'
that of the sun, is ellipiiL-, l)ut loii-
sidorably more ecceutnc ; the ilrsl
approximation to its orbit, 205.
Molioiiti of tlie nodes of, 20j. Oc-
ciiltations of, 21)7. Piiases of, 21 1.
It.s synodical periods, 212. Revo-
lutions of the apsides of, 210.
Physical constitution of, 217. lis
niountains, 218. Its aanuspliere,

219. Rotation of; libration of

220. Diminution of gravity at tiie ;
distance of it from the earth, 224.
Its gravity towards tlie earih; to-
wards the sun, 273. Its motion
disturbed by the sun's atlraclion,

332. Acceleration of its moan
motion; accounted for by Laplace,

333. Motion, parallactic, 18. Ap-
pearances resulting from diurnal
motion, 19. Real and apparent
motion of the earth described, HJ5.
Of bodies, 222. Laws of elliptic
motion, 226. Orbit of the earth
round the sun in accordance with
these law s, 227.

Mural circle, 89.

N.

Nebulte, Sir W. Herschel'.-; disco-
veries oi; 375. Resolvable, 376.
Aimular, 3'/ 8. Planetary, 378.

JNewton, his law of universal gravi-
tation, 225.

JVodes, their motion, 302.

JMutation, its jjhysical causes, 313.

O.

Olbers, Dr., 202.

Orbits, variation of their inclinations,
306.

P.

Parallax, 52.

Pendulum, 122.

Perturbations, 294. Of the planeta-
ry orbits, 319.

Planet, method of ascertaining its
mass, compared with that of the
sun, when it has a satellite, 274.

Planets, the, 231. Apparent motion
of, 2 J2. Their stations and retro-
gradations, 233. The sun their
natural centre of motion, 234.
Their apparent diameters and dis-

tances irom the sun, 235. Motions
of llu; iidbrior planets ; transits of,
236. Elongations of, 238. Their
sidereal periods, 240. Synodical
revolutions of, 241. Phases of
Mercury and Mars, 242. Transiis
oi" Venus explained, 243. Supe-
rior planets, 246. Their distances
and periods, 247. Method for de-
termining tlieir sidereal periods
and distances, 248. Elliptic ele-
ments of the planetary orbits, 251.
Their heliocentric and geocentric
places, 258. The four ultra-zodi-
acal planets, discovered in 1801,
201. The physical peculiarities,
and prol'.able condition of the
several planets, 202. Their u(>
parent and real diameters, 205.
Their periods unalterable, 335.
Their masses discovered inde-
pendently ol' satellites, 347.

Polar and horizontal points, 90.

Pole star, 46. Situation of 89.

Precession, its physical causes, 309.

Projectiles, motion of, 222. Curvili-
near path of, 222.

R.

Rays of light, refraction of, 3L

Keilecling circle, 103.

llellectlon, angle of, equal to the of
incidence, 91.

Refraction, 31. Of the atmosphere
31. Effects of, to raise all the
heavenly bodies higher above the
horizon in apjiearance than they
are in re:ilitv, 32. General notions
of its amount, and law of variation,
34. Terrestrial refraction, 38
Celestial refraction, 38.

Repetition, jiriaciple of, invented by
Borda, 103.

S.

Satellites, 272. Their motions round
their primary analogous to those
of the latter round the sun, 274.
Of Jupiter, 275. Their masses, 348.

Saturn, his satellites, 281.

Sea, action of the on the land, 117.

Seasons, explanation of the, 186.

Sextant and reflecting circle, lOL
Its optical property, 102.

.^iderenl clock, (i2.

Sidejeal year, 158.

396

INDEX.

Sidereal time, reckoned by the di-
urnal motion of tlie sturs, 02.

Sirius, its intrinsic brilliancy, 355.

Solar eclipses, 208. System, 231.

Sphere, celestial, 39. Projections
of, 146.

Stars, 52. Distance of, from the
earth, 53. Sidereal time reckoned
by the diurnal motion of the, 62.
Visible by day, 65. Fixed and
erratic, 155. Their relative mag-
nitude ; infinite number, 349.
Their distribution in the heavens,
351. Their distances, 352. The
centres of planetary systems, 356.
Periodical, 356. Temporary, 358.
Double, 360. Binary, 364. Their
orbits elliptic, 365. Their colours,
368. Their proper motions, 369.
Clusters of, 373. Globular clus-
ters of, 374. Irregular clusters of,
375. Nebulous, 377.

Sun, apparent motion of the, not
uniform, 176. Its apparent diame-
ter also variable, 177. Its orbit
not circular, but elliptical, 177.
Variation of ils distance, 179. Its
apparent annual motion, 180.
Parallax of, 180. Its distance and
magnitude, 183. Dimensions and
rotation of, 184. Mean and true
longitude of, 192. Equation of its
centre, 193. Phy.sical conslitulion
of, 197. Density of; force of gra-
vity on its surface, 227. The dis-
turbing eflect of, on the moon's
motion, 228.

Table, exhibiting degrees in differ-
ent latitudes, expressed in British

standard feet, as resulting from
actual measurement, 111.

Telescope, 85. Application of, the
grand source of all the precision
of modern astronomy, 86. Differ-
ences of dechnation measured by,
87.

Terrestrial refraction, 38.

Theodolite, construction of the, 144.

Tides, their physical cause, 314.

Time, mea.su rement of, 78. Its
measures, 381.

Trade-winds, 124, Explanation of
this phenomenon, 125. Compen-
sation of, 127.

Transit instrument, 76.

Trignometriftal survey, 142.

Tropical and anomalistic years, 195.

Twilight caused by the reflection
of the sun and the moon on the
atmosphere, 35.

U.

IJranographical problems, 173.
Uranography, 151.
Uranus, his satellites, 282.

V.

Varialions, periodic aaid secular,
320,

Year, tro])ical, the civil measure of
time, 381. Sidereal, 381.

Zodiac, the, 157.
Zodiacal light, 380.

THE END.

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